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The Twentieth Century 
Arithmetic. 


Part II. 

FOR 

GRAMMAR SCHOOLS. 


189b 

a**' ' 

B.L., 

Graduate U. S. Naval Academy. 



Atlanta, Ga. : 

The Foote & Davies Company. 
1896. 





Copyright, 1896, 

By ERNEST E. WEST. 


PREFACE. 


T HE TWENTIETH CENTURY ARITHMETIC is the result of 
labor expended from time to time over a period of six years, 
though the whole work was revised and unified just prior to pub¬ 
lication. During those years, the World’s Fair at Chicago and the 
Cotton States and International Exposition at Atlanta, were held, and 
the famous Report of the Committee of Ten was issued. At the Fair 
and at the Exposition, I made an exhaustive examination of the dis¬ 
plays of work in arithmetic. Having availed myself of the knowledge 
there obtained, and having adopted some of the recommendations of 
the Committee of Ten, I present this volume to pupils and teachers 
with the confidence which such a backing warrants. I add to this the 
confidence of a successful personal test in the schoolroom, during 
these six years, of many of the methods and principles which are 
used in this book. 

The problems involve geographical/historical, physical, and math¬ 
ematical constants to an extent not found in any other arithmetic. 
This correlation of arithmetic with other studies, adds to it a value 
not easily estimated. 

Such subjects as duodecimals, banking, insurance, and exchange 
have been omitted. To teachers who have kept in touch with 
modern educational thought, these omissions need no defense. Suf¬ 
fice it to say that it is not the province of arithmetic to teach all its 
applications. 

On page 37, et seq., will be found a subject which, if thoroughly 
learned by the pupil, will, I believe, do more towards making him a 
good mathematician than the same amount of time spent on any 
other part of arithmetic. 

I call particular attention to my treatment of Denominate Num¬ 
bers, Proportion, and Interest. 

The forms for solutions throughout the book are believed to be the 
best ever presented. I advise a strict adherence to them. In almost 
all of these forms, all the work necessary to the solution is so ar¬ 
ranged that the pupil can detect his own error without duplicating 
his work, and the teacher can correct an examination paper with 
delight—if such a thing is possible. 

Answers will be inserted under same cover with the arithmetic in 
future editions. They are furnished free with each volume of this 
edition. ERNEST E. WEST. 

Atlanta, Ga., August 16 , 1896. 



ERRATA. 


PAGE 

ARTICLE. 

EXAMPLE. 

80 

145 

b. 

A # 

should be —5——^— 

8 f 

should begin 9^— 

81 

145 

26. 

48 

154 

18. 

555 should be 185 

62 

184 

5. 

— should be ; 

62 

184 

9. 

fff should be fff 

82 

218 

In Analysis. should be V 

84 

215 

12. 

gal. should be bu. 

125 

255 

b. 

90° should be 122° 

144 

287 

20. 

27 should be 2.7 


Make these corrections at the proper places before using the book. 





CONTENTS OF PART II. 


PAGE. 


Fractions. 1 

Reduction. 7 

Review.. . 13 

Addition.14 

Subtraction.18 

Multiplication. 22 

Division.25 

Review.31 

Equations. 37 

Examples in Fractions . . 43 

Decimals . .. 47 

Addition.50 

Subtraction.51 

Multiplication.51 

Contracted Multiplication . 54 

Division.56 

Contracted Division . . 60 

Review.62 

United States Money ... 63 

Aliquot Parts. 68 

Review. 73 

Denominate Numbers 75 

I. Linear Measure ... 76 

II. Avoirdupois Weight 76 

III. Liquid Measure ... 77 

IV. Dry Measure .... 77 

Reduction.78 

Addition.84 

Subtraction.85 

Multiplication. 86 

Division.87 

Review. 88 

V. Troy Weight .... 89 

VI. Apothecaries’ Weight . 91 


PAGE. 


VII. Numbers. 92 

VIII. Paper.92 

IX. English Money ... 93 

X. Square Measure ... 96 

XI. Cubic Measure . . . 100 

XII. Wood Measure . . . 102 

XIII. Circular Measure . 103 

XIV. Time. 105 


Exact Difference in Days . 109 
Longitude and Time . . .110 

Time at Different Places 113 
Finding Longitude at Sea 114 


Distance and Time . . .115 

Work and Time .... 122 
Measurement of Tempera¬ 
ture . 124 

Review.128 

The Metric System .... 131 

Notation.132 

Numeration.134 

Reduction.135 

Fundamental Operations . 136 
Square Measure . . . .138 

Land Measure.139 

Cubic Measure. 139 

Wood Measure. 139 

Comparison with U. S. 

Measures. 140 

Specific Gravity .... 142 

Review.144 

Ratio . % '. 146 

Reduction.147 

Direct Variation .... 150 
Inverse Variation .... 152 



































VI 


CONTENTS, 


PAGE. 


Simple Proportion .... 154 

Problems.156 

Joint Variation .... 161 
Compound Proportion .163 

Problems.165 

Partitive Proportion . . . 169 

Comparison of Numbers . . 172 

Arithmetical Ratio . . . .174 

Percentage.176 

To Find the Percentage . 179 
To Find the Rate .... 179 
To Find the Base .... 180 
Problems Involving Amount 181 
Problems Involving Differ¬ 
ence ..181 

Miscellaneous Problems . 182 

Gain and Loss.184 

Miscellaneous Problems . 186 
Simple Interest.189 


PAGE. 


6 % Method.191 

Exact Interest.193 

To Find the Rate .... 195 
To Find the Time . . . .196 

To Find the Principal . . 198 

Problems Involving Amount 199 
Promissory Notes .... 200 

Annual Interest.204 

Compound Interest.... 206 

Bank Discount.207 

True Discount.210 

Partial Payments .... 212 

Involution.220 

Evolution.220 

Square Root.221 

Cube Root.226 

Formulas.228 

Progressions.232 





















The Twentieth Century 
Arithmetic. 


Part II. 


FRACTIONS. 


103. 




1. These circles are divided into equal parts. 



2. What part of A is w ? 

3. What part of B is x ? 

4. What part of C is y ? 

5. What part of D is z ? 


Ans. One half. 

Ans. One third. J. 
Ans. One fourth. \ 
Ans. One fifth. 









9 


THE 20th CENTUR.Y ARITHMETIC. 


What is each part called when a whole is divided into 

6 . 6 equal parts ? 9 equal parts ? 12 equal parts ? 

7. 7 equal parts ? 10 equal parts ? 13 equal parts ? 

8. 8 equal parts ? 11 equal parts ? 14 equal parts ? 

9. Two parts, each the size of x, would be what part of 
the whole ? 

10. Three parts, each the size of y, would be what part of 
the whole ? 

11. Two parts, each the size of z, would be what part of 
the whole ? Three parts ? Four parts ? 

12. How many halves does it take to make a whole ? 

13. How many thirds does it take to make a whole ? 

14. How many fourths does it take to make a whole ? 

15. How many fifths does it take to make a whole ? 


104. 




1. What part of the whole is A ? 

2. What part of the whole is B ? 

3 . What part of the whole is C ? 

4 . What part of the whole is D ? 

5 . Which is the larger, \ or f ? 5 or j ? 






FRACTIONS. 


3 


105. 


One Whole. 


A 



2 

3 

4 

C 1 


3 

- 

5 

—- - — 1 — 

7 

-Q- -J 

DlliRi 

3 | 4 

-JL-ISi 

X lJiS ! 'ii] 

9 

11 liHgi 

13 

15 lllsini 


Let the pupil write the answers to these questions, numbering his 
answers to correspond with the questions. 

1. How many parts in A ? In B ? In C ? In D ? 

2. How many sixteenths make a whole ? 

3. What part of the whole is 5 parts of C ? 7 parts of D ? 

4. Express in figures 3 parts of B. 7 of C. 11 of D. 

5. Are five eighths equal to eleven sixteenths ? 

6. How many parts of B equal one of A ? Of D equal 
one of C ? 

7. Which is smaller, three eighths or five sixteenths ? 

8. Which is larger, 9 parts of D or 5 of C ? 3 of B or 7 
of C? 

9 . Is it true that £ equals || ? That f equals | ? 

10. How many sixteenths make \ ? How many fourths 
make \ ? 

11. What part of the whole is 7 parts of C ? 3 parts of B ? 

12. How many are 3 eighths and 4 eighths ? f-j-f ? 

13. How many are 2 eighths from 7 eighths ? f—§ ? 

14. How much is 3 times three sixteenths ? 3X fV ? 

15. What is one half of 12 sixteenths ? Jf-r-2 ? 


DEFINITIONS. 

106. One or more equal parts of a unit is called a 
fraction; as, one fourth; three fourths. 
















4 


THE 20th CENTURY ARITHMETIC . 


A fraction may be expressed by words or by figures : 

One third=4 Three fourths=f Seven eighths=| 
Two thirds=§ Three fifths=f Nine sixteenths= T % 

a. The number above the line is called the numerator. 

It shows how many parts are taken. 

b. The number below the line is called the denominator. 
It shows into how many equal parts the whole is divided. It also 

names the parts; in two thirds, thirds is a name , just as boys is in 
two boys. 

C. One of the parts into which the unit is divided is called 

the fractional unit. 

In f, ^ is the fractional unit. 

In -f, which number expresses the size of the parts ? The 
number of parts ? What is the fractional unit ? What is 
the fractional unit of $ ? f ? T 4 T ? t 9 q ? f ? 

d. The terms of a fraction are the numerator and denomi¬ 
nator. 

f. Three fifths. 3 and 5 are the terms. 3 is the num- 


erator and 5 the denominator. 

The fractional unit is 

Analyze in this ivay : 






a. b. c. 

d. 

w. 

X. 

y- 

Z. 

% % */ 

i % 

T3 

1 3 

2 7 

a 

T 2 4 2 5 

«• % % ’/ 

i % 

t 6 t 

n 


tVtt 

*■ % % “/ 

i % 

T4 

H 


hn 

When both terms of a 

fraction 

are less 

than 

9, the 

line between 


them may be made sloping; otherwise, horizontal. 


107. Write with figures : 

1. Four fifths. 

2. Seven eighths. 

3. Five thirteenths. 

4. Nine twentieths. 


5. Seventeen thirty-fourths. 

6. Eleven twenty-ninths. 

7. Twelve forty-thirds. 

8. Nineteen seventy-firsts. 



FRACTIONS. 


5 


9 . Seven eighty-fourths. 

10. Twenty-three forty-sevenths. 

11. Thirty-five thirty-sevenths. 

12. Seventy ninety-seconds. 

13 . 40 one-hundred-twenty-thirds. 

14 . Eighty two-hundred-sixteenths. 

15 . Ninety-four ninety-fifths. 

16 . Seventeen ninetieths. 

17 . 23 four-hundred-fourths. 

18 . Twenty-three divided by 404. 

19 . 41 fifty-thirds. 

20. 35 forty-sevenths. 

21. 67 ninetieths. 

22. 82 eighty-sevenths. 


23 . 

17 fortieths. 

24 . 

95 ninety-ninths. 

25 . 

87 ninety-thirds. 

26 . 

132 

divided by 145. 

27 . 

284 

divided by 345. 

28 . 

136 

divided by 175. 

29 . 

241 

divided by 287. 

30 . 

362 

divided by 584. 

31 . 

25^ 

-55. 

32 . 

36-r 

-42. 

33 . 

48- 

-56. 

34 . 

124- 

--146. 

35 . 

63—- 

-99. 

36 . 

89-- 

143. 


108 . 


A 


B 


C 


-1-i-1-f-1- 

- 1 - 1 - 

The distance A C represents 2 units. 

It may be seen that the length A B is 2 thirds of 1 unit, 
or 1 third of 2 units. The fraction is also an expression of 
division in which 2 is the dividend, 3 the divisor, and § 
itself the quotient. 

2 = | 0 f i = i 0 f 2=2-^3 
Analyze in this way : 



a. 

b. 

c. 

d. 

w. 

X. 

y. 

z. 

l. 

§ 

t 

3 

¥ 

I 

i 

f 

T°T 

2. 

¥ 

¥ 

1 

A 

5 

¥ 


TT 

¥ 

3. 

¥ 

A 

3 

7 

1 

t 

A 

H 

if 






G 


THE 20th CENTURY ARITHMETIC. 


109. DEFINITIONS. 

. . . . , ( Common or ( Simple or ( Proper or 

A fraction is either j Deoimal) \ Complex, I Improper. 

a. A decimal fraction is one whose denominator is 10 or 
a power of 10. (Art. 162.) 

b. A fraction with any other denominator is a common 
fraction. 

C. A simple fraction is one having integers for both 
terms: -J, f. 

d. A complex fraction is one having a fraction in one 

2 7 1 

or both terms : 4|, y. 

e. A proper fraction is one whose numerator is less than 

the denominator : f. 

f. An improper fraction is one whose numerator equals 

or is greater than the denominator : f. 

The value (Art. 117) of a proper fraction is less than one whole. 
The value of an improper fraction equals or is greater than one 
whole, and, hence, is not properly part of one whole. 

g\ A mixed number is an integer and a fraction : 5J, 4^. 
Say and between the integer and fraction. 

110. What kind of fraction is £ ? Ans. Common, 
simple, and proper. 

What kind of a fraction is 



a. 

b. 

c. 

d. 

e. 

1. 

7 ? 

TT) * 

+ ? 

ToV ? 

W? 

A? 

2. 

11? 

'■5 * 


r? 

5J ? 

2 

t4h ? 

3. 

-f- ? 

y 

r? 

i? 

IL? 
1 0 0 

4. 

H? 

i? 

9 xl ? 

1 0 0 

? 

V? 

5. 

44 ? 

7 

°4 

8 ? 
TO OJJ 1 


44 ? 
14 



FRACTIONS . 


7 


6. Write a proper and an improper fraction with the 

figures 5 and 11. 4 and 9. 6 and 18. 

7. Write a mixed number and a complex fraction with 
the figures 2, 8 and 5. 3, 7 and 11. 

8. Write a proper and an improper fraction with 8 and 
7. 4 and 5. 7 and 12. 


REDUCTION. 

111. To reduce a fraction is to change its form without 
changing its value. 

a. A fraction is in its simplest form when the terms are 
integers and prime to each other. 

b. 2 boys-f 3 girls do not make 5 boys or 5 girls; but 2 boys 
are 2 children and 3 girls are 3 children, and 2 children-b3 
children make 5 children. 

We cannot add things with different names, therefore we 
cannot add 2 thirds and 3 fourths. But just as boys and 
girls are both children, we can find a common name for 
thirds and fourths and then add. This process of changing 
fractions to a common name is a process in reduction. 


An Integer to a Fraction. 

112. If you divide one apple into a number of equal parts 
and keep all the parts, you still have one apple. Expressed 
mathematically, § or f or § of an apple is all of it. 

§=1. f=l. |=1. 

How many thirds in 4 apples ? 

Analysis.— In one apple there are three thirds; then in 4 apples 
there are 4 times 3, or 12 thirds. Ans. - 1 




8 


THE mh CENTURY ARITHMETIC. 


1. How many thirds in 5 apples ? 

2. How many halves in 8 apples ? 

3. How many fourths in 7 apples ? 


4. How many fifths in 8 apples ? 

5. How many sevenths in 4 apples ? 


Reduce 


6 . 

7. 

8 . 

9. 

10 . 
11 . 
12 . 

13. 

14. 

15. 


8 to a fraction whose denominator is 5. 
2 to a fraction whose denominator is 8. 

4 to a fraction whose denominator is 7. 

5 to a fraction whose denominator is 8. 
7 to a fraction whose denominator is 4. 


7 to thirds. 

8 to halves. 

9 to fourths. 

11 to sixths. 

18 to sevenths. 


16. 

17. 

18. 

19. 

20 . 


4 to ninths. 

7 to fourths. 
9 to eighths. 
6 to sixths. 

8 to thirds. 


21. 9 to elevenths. 

22. 11 to twentieths. 

23. 17 to fourteenths. 

24. 4 to forty-firsts. 

25. 18 to ninetieths. 


Mixed Number to a Fraction. 

113. Reduce 4§ to a fraction. 

4=12 thirds. 12 thirds-f-2 thirds=^. 
Directions.— Multiply the denominator of the fraction by the in¬ 
teger; to the product add the numerator of the fraction and place 
the sum over the denominator. 


Reduce to a fraction : 


1 . 3 £. 

2. 4f. 

3. 7f. 

4. Hf. 

5. 18}. 

6. 15#. 


7. 20}. 

8. 42 t \ 

9. 61}J 

10. 78 

11 . 85}f 

12. 91X 


13. 

14. 

15. 

16. 

17. 

18. 


1^1 T ^. 

216}. 

fAV 


82 


if 0 




For other examples use answers to those in Art. 114 


19. 48}f. 

20 . 97*. 

21. 19}}. 

22. 40*. 

23. 76}*. 

24. 64ff. 



FRACTIONS. 


V 


Improper Fraction to an Integer or Mixed Number. 

114. Reduce J g* to a mixed number. 

Analysis. —Since 3 thirds make 1, 14 thirds are as many l’s as 
there are 3’s in 14, which is 4 and 2 thirds. Arts. 4§. 

Directions. —Divide the numerator by the denominator. 

Reduce to a ivhole or mixed number: 


1. 

9 

2 ^ 

6 . y. 

11. 

'W'- 

16. 

4 §t 4 - 

21. 

w- 

2. 

¥• 

7. W- 

12. 

W-. 

17. 

-4t a - 

22. 

lys 

3. 

¥• 

8. -W- 

13. 

W- 

18. 

W- 

10 

GO 


4. 

¥• 

9. Hfi. 

14. 

-t!-* 

19. 

-4I-- 

24. 

-1.77 A 
J 5 

5. 

¥• 

10. W- 

15. 


20. 


25. 



For other examples use answers to those in Art. 113. 



Reduction to Higher Terms. 


116. 

Examine the following : 



1 i 2 

. 3 


i 

, 2 , 8 , 

4 S 

0 

1 2 

-i 8 i 4 i 8 i 0 i 

7 8 . 9 . lO . 11 12 



/W 2 


/ r 

. c= ( 1/eN 

/ 1 /^\\ 



l * s' 

\. \ V12 


v ]/ 

4 

- CM 1 

\X / 

C\ 

y\^> 

M/ 


§ — I — 

2X2 4 2X4 8 

3X2” 6 3x4 12 


Multiplying both terms of a fraction by the same number does 
not change the value (Art. 117) of the fraction. 














10 


THE mh CENTURY ARITHMETIC. 


This is true because § and f are each equal to 1, and when 
§ is multiplied by what is equal to 1 , the product is equal 
to §. 

Reduce § to 12ths. 


1=41 Directions. —Divide the new denominator by the 

4=y\ denominator of the given fraction and multiply 

f=y 8 o Ans. both terms of the given fraction by the quotient. 


Reduce 

1. § to 12ths. 6. f 

2. | to 16ths. 7 . f 

. 3. f to 8ths. 8. | 

4. f to 21sts. 9 . | 

5. T \ to 22ds. io. | 

16. §, and f to 18ths. 

17. 4> f ? and y 7 y to 12ths. 

18. % , f, and t 5 5 to 14ths. 

19. f, f, and t 3 o to 90ths. 

20. $, I, and \ to 20ths. 


to 18ths. ii. 23 - to lOOths. 

to 10 ths. 12 . fV to T 2 ds. 

to 24ths. 13 . 44 to lOOths. 

to 40ths. 14 . y 4 y to 65ths. 

to lOtlis. 15. to 64ths. 

21 . 4? f, and 4 to 24ths. 


22 . 4 , T V, and 4 to 15ths. 

23. |, y\, and x \ to 48ths. 

24. |, r 7 ¥ , and ^ to 72ds. 

25. T 5 2, T V, and 4V to 60ths. 
A, and to lOOths. 

27. f, 44? 4§> I? and 44 to lOOths. 


116. The least denominator to which several fractions 
can be reduced must be a number divisible by all the de¬ 
nominators (see Directions in Art. 115); it is, therefore, the 
L. C. M. of the denominators. 

Fractions have a common denominator when their de¬ 
nominators are the same number : 4 ? t? f- 


117 . The value of a fraction depends on the relative mag¬ 
nitude of its terms : § is greater than In order to com¬ 
pare the values of fractions, it is often necessary to reduce 
them to a common denominator. For this purpose, the 
least common denominator is to be preferred. 




FRACTIONS. 


11 


118 . We know that f is greater than f, or that f is 
greater than T 3 T ; but is § or j 5 % the greater ? We can best 
compare their values by reducing them to the same denomi¬ 
nator: 86 is the L. C. M. of 9 and 12. 

5 15. therefore, f is the greater. 


tf —\ 


Which is the greater , 

1 . 

2 . 

3. 


for |? 


? 


TF 0r if ? 

H or if ? 


6 . 

7. 

8 . 

9. 

10 . 


¥ or fff ? 
tf °r ff ? 
f or if ? 
t or t 7 f ? 
if Or ri ? 


11. I or |f ? 

12. | or if ? 

13. y 5 ^ or T 7 ^ ? 

14. y 7 y or if ? 

15. H or f ? 


119 . Arrange in their order of magnitude {greatest first): 


1. 

1 3 

F * TF * 

FT • 

8. 

8 

3+1 

8- 

-1 

2. 

1 ft 

F* TF* 

fV 

T’ 

4+1’ 

4- 

-l' 

3. 

i* if* 

f T • 


9 

9+1 

9- 

-1 

4. 

f* if* 

Si- 

9. 

18’ 

11 

18+1’ 

11+1 

18- 

11- 

-1 

-1 

5. 

T* FT* 

A- 

10. 

IT 

15+1’ 

15- 

-1 

6. 

7 7 

TF* F* 

Hi A- 

11. 

7 

7+1 

7- 

-1 

7. 

F* TF* 

FF* if- 

18’ 

18+1’ 

18- 

-1 


Reducing a Fraction to Lower Terms. 

120 . Since f=y 8 2 - (Art. 115), t 8 2 == §* 

Notice that if you divide both terms of T 8 y by 4, you obtain 
§. Hence, 

If both terms of a fraction be divided by the same number , the 
value of the fraction is not changed. 

A fraction is in its lowest terms when the terms have no 
common factor (Art. Ill, a.). Hence, if we divide the 
terms of a fraction by the greatest number possible (their 
G. C. D.), the fraction will at once be reduced to its lowest 











12 


THE 20th CENTURY ARITHMETIC . 


terms. Or, if we divide separately by factors common to the 
terms till the terms have no common factor, we obtain the 
same result. 


FIRST METHOD. 

131 . Reduce Iff to its lowest terms. 

Solution.—D ividing each term by 5 we get Dividing each 

term of by 3 we get tt- Therefore, Ttl == §i == T 7 T- 

The rules for determining by inspection when a number is 
divisible by 2, 8, 5 and 11, should be used. 


Reduce to lowest terms: 


1. 

3 0 

4F* 

6. 

m- 

11. 

tfW 

16. 

m- 

2. 

ff. 

7. 

tWV- 

12. 


17. 

me- 

3. 

t¥V- 

8. 

«**• 

13. 

19 8 

T2FF* 

18. 

TAsV 

4. 

tu¬ 

9. 

tWs 0 o • 

14. 

mt- 

19. 

fVit 4 ? • 

5. 

rn- 

10. 

118 8 

TFT) FIT* 

15. 

mu- 

20. 



CANCELLATION. 

122 . If the terms consist of factors, equal factors may 
be disposed of by a process called cancellation. Cancella¬ 
tion is the process of dividing both > terms of a fraction by 
the same number (a common factor). 

a. 2X^X0 2 a. Divide both terms by 3 and by 5. 

$X^X7 7 

b. 7 into 14, 2 times ; 2 into 8, 4 times ; 

b. ^X0XX^ 1 3 into 15, 5 times; cancel the 5’s. 

" JySyJW C ° Py thiS example ' 

/' A p A The divisions are indicated as shown in 

^ ^ a and b, by cancelling the equal factors. 






.v»? A(: . ((h\ r S. 


18 


What is the value of 


1. 

3X5 

5X7 1 

6. 

68X28X14 n 

11. 

30X28X 45 „ 

49X26X21 ? 

42X44X105 ? 

2. 

4X5 . 

8X8 * 

7. 

56 X 24 X 9 „ 
21X27X16 * 

12. 

45X525X86 

975X105X48 

3. 

8X5X 7 o 

8. 

48X26X19 o 

13. 

180 X 26 X 72 „ 

2X5X18 1 

38X52X16 ' 

89X84X99 ? 

4. 

15X8X12 

9. 

25X 34X18 „ 
51X45X30 ' 

14. 

43X82X525 o 

18X6X 5 - ‘ 

16X86X975 ? 

5. 

9X 7X35 „ 

10. 

18X16X23 „ 

15. 

132X105X35 4 

49X15X11 ^ 

72 X 92 X 5 ? 

84X180X49 


Reduction to Lowest Terms. 

SECOND METHOD. 

123 . Reduce to its lowest terms. 

Solution. —The G. C. D. of 133 and 190 is 19. Dividing 133 and 180 
by 19, we get respectively 7 and 10; therefore, it o '—to • A'is. 

Directions. —Find the G. C. D. of numerator and denominator. 
Divide the terms by the G. C. D. 

Reduce to lowest terms : 


1. 

m- 

7. 


13. 

9 9 0 

19. 

itif • 

25. 

5 9 8 5 

2. 

133 

8. 

Hf- 

14. 

8 0 8 5 

T 

20. 

mi 

26. 

IMt 

3. 

m- 

9. 


15. 

93 50 
T3?T75- 

21. 

40 8 7 
ffSTTT- 

27. 

4 5 7 8 

4. 

tVA- 

10. 

«#*• 

16. 

3 5 5 5 

22. 

tWj- 

28. 

59 8 6 
W8W7 

5. 

tWt- 

11. 

AA- 

17. 

3 5 5 5 

23. 

12 43 
T4ffS' 

29. 

9 1 5 
TO^S 

6. 

m- 

12. 

t¥tV 

18. 


24. 

3 8 50 
T7JTV 

30. T Ws 


124 . REVIEW. 

1. What is a fraction ? 

2. Name the terms of the fraction f. 

3. What is a fractional unit ? 


















14 


THE 20tli CENTURY ARITHMETIC. 


4. Write the fraction eighteen thirty-firsts. 

5. What three facts does §-express ? (Art. 108). 

6. What kind of a fraction is f ? Simple, etc. 

7. Reduce 7 to fifths. 

8. Reduce 8f to a fraction. 

9. Reduce \ 7 - to a mixed number. 

10. Reduce f to a fraction whose denominator is 68. 

11. Arrange in their order of magnitude T 4 oV, and 

12. Reduce to its lowest terms, using factors. 


13. Reduce 


14X88X110X15 
42X17X 55X 8 


to a fraction in lowest terms. 


14. Reduce to lowest terms, using G. C. D. 

15. Reduce f, f, £, and to equal fractions with their 
least common numerator, then arrange them in their order 
of magnitude. 


16. Arrange in their order of magnitude f , T \, and ff. 

17. Find a fraction intermediate in value to f and t 8 t, 
whose denominator is 88. 


18. Find a fraction intermediate in value to f and T 9 ^. 

19. Add f and 4, giving the answer as an improper frac¬ 
tion. What is this operation ? (Art. 118). 

20. Reduce to lowest terms, using factors. 


ADDITION OF FRACTIONS. 

125 . Addition is the process of uniting two or more like 
quantities into one quantity. Quantities are alike when 
they have the same name. Since the denominator of a frac¬ 
tion names the parts, fractions can be added only when they 
have the same denominator. 

2 boys -f8 boys +5 boys =10 boys. 

2 fourths-f-3 fourths-f 5 fourths=10 fourths. 

1+14-1 =¥-=2i. 




FRACTIONS. 


15 



Directions.— Add the numerators and place the sum over the 
common denominator. If the sum is an improper fraction, reduce it 
to a mixed number. 

Find the value of 


1. 


+ f 

+ #• 

11. 


2. 

§ 

+ 1 

+ i • 

12. 

iff iff Af A* 

3. 


+ 1 

4-f • 

13. 

A f Af Afli- 

4. 

i 

1 2 

5 

.+ *■ 

14. 

Af iff iff if- 

5. 

T 4 2 

_ 1 _ 9 
TT2 

+A- 

15. 

2f+ 3f+ll|. 

6. 

A 

-1- 8 
TTJ 

+ tV 

16. 

9§f 4|+16f. 

7. 

11 
2 0 

+ 2 9 0 

+A- 

17. 

4*f9*f 18*. 

8. 

ft 

+A 

+A- 

18. 

6U+7H+1A- 

9. 

A 

4-H 

A* 

19. 

51f+ 1|+ 5J. 

10. 

it 

4" 2% 

+A- 

20. 

8|+ 9f+19|. 


The author suggests that pupils solve all examples, using the form 
given in the specimen example. The form given makes the solution 
the easiest possible. 

126. Before fractions can be added they must be re¬ 
duced to a common denominator. The least common de¬ 
nominator possible is the most convenient, because its use 
makes the least amount of work (Art. 116). The least 
denominator possible is the L. C. M. of the denominators. 

Add f, | and f. 

The L. C. M. of 3, 6,and 8 is 24. 








16 


THE 20th CENTURY ARITHMETIC. 


24ths. 

§=16 

f=20 

_t=_9 

^^=45 Ans. 
ff=X5=l|. 


Directions. —Find the L. C. M. of the de¬ 
nominators. Reduce each fraction to a frac¬ 
tion having thisL. C. M. for its denominator. 
Add the resulting fractions. 

Arrange the work as shown. 3 into 24, 8; 
twice 8=16, etc. 


Find the value of 

1. i+i+f- 16. f+j+i. 31. «+§+§ + §. 

2- i+i+ii- 17. t+A+i- 32. i+i+i+H- 

3. f+i+A- 18. S+A+A- 33. A+A+A+H- 

4. t + f+A- 19. f+if+H- 34. f + |+i+ii. 

5. f+f+A- 20. A+A+A- 35. #§+§+A+V-- 

6. §+§+A- 21. f+A+ix- 36. f+A+H+A- 

7- 4+i+if- 22. §+ T \+i. 37. *+§* + §§+}§. 

8. ♦+§+§§. 23. V-+A+A- 38. A+A\+A^+tUU- 

9- I+4+A- 24. TT+l+xfx- 39. A+x 7 A+A 8 o 4 <r+xMA- 

10. f+A+§*. 25. x 7 TT+H+tU- 40. A+x 5 x + l#+l+§|. 

11. §+§+§• 26. «+«+§§*• 41. A+H+*+t+§. 

12. i+f+A- 27. A+H+fi- 42. H+i+il+4+M- 

13. 4+f+A- 28. A+H+M- 43. M+f+f+H+ff- - 

14. f+S+A- 29. A+H+H. 44. H+l+H+H+f. 

15. §+$+§. 30. A+H+H- 45. §+«+§§+§§+§*. 

For other examples, use those in Art. 115. 


127. Addition of Mixed Numbers. 

What is the sum of 4f, 2f, and 5§ ? 

24ths. 

4», 16 

2 |, 20 

5f, 9 Directions. —Add the fractions, then 

the integers. 

11 Arrange the work as shown. 


12$. Ans. 








FRACTIONS. 


17 


Find the value of 


1. 

3i+4|+7f. 

11. 

2. 

8f+9i+2}. 

12. 

3. 

7|+2|+5J. 

13. 

4. 

7i+5J+2. 

14. 

5. 

4f+5f+8 t V 

15. 

6. 

9|+7|+6^. 

16. 

7. 

5J+lf+7iJ. 

17. 

8. 

6f+2i+|f. 

18. 

9. 

7f+5J+llJJ. 

19. 

10. 

4ft+6*+19,V 

20. 


53£+26f ■13 t 5 ¥ . 

17*+5i+28*. 

84*+82A+87*: 

13»+16*+17f+50f. 

5f+8|-|-#-j-2$-j-lh£. 

126f+81H+78*+16H. 

71+841f+28#+44j+10f. 

32 fx +4113^--j-78#f. 

or++17 T V(j +1 9 tH o • 

SH-®*+Wfr+4f. 


128. Miscellaneous Examples. 

1. Which is the greater, y 2 - or fff ?. 

2 . Which is the greater, I, or ? 

3. Arrange in order of magnitude, &, T \, and i+A- 

g 5_j_ ^ pj_^ 

4. Reduce—, ^—j to equal fractions with their 

least common numerator, then compare their values. 

5. Reduce iff#, tVfW? fIMt to their lowest terms. 

6. Find a mixed number intermediate in value to \y- 
and 8f. 

7. A thermometer registered 35| degrees above zero in 
the morning. By noon it had risen 21J degrees. What did 
it register then ? 

8. Five bolts of calico contain respectively 35#, 32J, 31#, 
32#, and 30f yards. How much is there altogether ? 

9. Ella spent $2j for Christmas presents and had $3 T 7 ^ 
left. How much did she have at first ? 

10. -ft- of every rainbow is yellow, & is orange, and # is 
red. What part of the whole rainbow are these three colors ? 




18 


THE 20th CENTURY ARITHMETIC. 


11 . In 1895, X 3 T of the population of the United States 
was living in New York State, yVm Pennsylvania, in Ill¬ 
inois, in Ohio, and ^ in Missouri. What part of the 
total population lived in these five States ? 

12 . Mary’s school time is taken up as follows : T % of the 
time by language lessons, £ by arithmetic, and x 2 g by geog¬ 
raphy. What part of her whole time is taken by these three 
studies ? 

13. In 1894, of the total expenditure of the United 
States government was for pensions, ^ for the navy de¬ 
partment, 3 % for the war department, and y 1 ^ for interest 
on the public debt. What part of the total expenditure was 
the sum of these four items ? 


SUBTRACTION OF FRACTIONS. 


129. Subtraction consists in expressing by one quantity 
the difference between two like quantities. 

Fractions can be subtracted only when they have the same 
denominator. 


f—f=f. 


What is the value of 



130. When necessary, reduce the fractions to a common 
denominator. The least common denominator is to be pre¬ 
ferred. 

What is the value of f—§ ? 


24ths. 


Directions. — Reduce the fractions to a 
common denominator. Subtract the numer¬ 
ators and place the difference over the com¬ 
mon denominator. 


1=20 
1 = 9 


Ans. 





FRACTIONS . 


19 


1. 

i—b 

11. 

1 2_8 

21. 

1 7_1 9 

O 1 

43_2 1 

49 SS- 

5 7 ' 

SS-SO- 

o 1. 

2. 

i — i - 

12. 

It— 1. 

22. 

it-H- 

32. 

33_ 7 

4 0 4 8- 

3. 

1—#• 

13. 

!#—ft. 

23. 

It — li¬ 

33. 

ft Is- 

4. 

i—h 

14. 

¥—?#■ 

24. 

lt if • 

34. 

2 8_ 7 

35-SO • 

5. 

V~2. 

15. 

II ft- 

25. 

It-lt- 

35. 

11—f- 

6. 

#—#• 

16. 

2—##. 

26. 

If—Il¬ 

36. 

is — II- 

7. 

t-l- 

17. 

ft 5 * T 2 ft 

. 27. 

ls— It- 

37. 

1 5_ 8 

" 8 " TS • 

8. 

io tV 

18. 

If—1- 

28. 

19_16 

2S T7 • 

38. 

it—If- 

9. 

H—t# 

19. 

¥—W-- 

29. 

T§—T§- 

39. 

TT2-TSS 

10. 

1 T2* 

20. 

«-«• 

30. 

if — I- 

40. 

ff-li- 

131 

1 

Subtraction of 

Mixed Numbers. 



Directions. —Subtract the fractions, then the integers. 


a. 7f—4f=what ? 

24ths. 

7f =7 20 
4j =4 9 

Ans. 3^=3 


11 Ans. 

The answers, 3^4 an ^ ai ' e written last. 

In b., 15 20ths can not be taken from 8 20ths 


b. 4f—lf=what ? 

20ths. 20ths. 

4f =4 8= 8 28 

If =1 15= 1 15 
2i§ = 2 18 


therefore, take 1, 


which equals 20 20ths, 

from 

4 and 

add it to 8 20ths. 



the value of 







1. 

•61—8#. 

11. 

Ilf— 

-6. 

21. 

1413 

I^t? 

-12#! 

2. 

8x5 * r, ft ■ 

12. 

53|§- 

-27 ft. 

22. 

11— 

6##. 

3. 

4f-2#. 

13. 

1 43/cr — 47f|. 

23. 

16— 

4f. 

4. 

Ilf—6*. 

14. 

24§— 

-Ilf. 

24. 

23— 

18#. 

5. 

3|—f. 

15. 

96#- 

-87##. 

25. 

20—10#§. 

6. 

«A—2|. 

16. 

16#- 

-8#. 

26. 

19#- 

-11. 

7. 

2#—If. 

17. 

17#- 

■7#. 

27. 

14ft- 

—10. 

8. 

18 J—5|J. 

18. 

16ft- 

-8«. 

28. 

13ft' 

—12. 

9. 

5—8f 

19. 

24||- 

—11##. 

29. 

16ft- 

—11. 

10. 

3ft— l|f • 

20. 

19#- 

•10H. 

30. 

14##- 

—12. 







20 


THE 20th CENTURY ARITHMETIC . 


Addition and Subtraction. 

132. Find the value of f-ff—f— t 7 ¥+S- 
The L. C. M. of all the denominators is 24. 


+ 24ths. — 24ths. 

1=16 I = 20 

4= 9 T V= 14 

1=18 34 

48 

— *=f. Ans. 


Directions. —Find the sum of all 
the + quantities and the sum of all 
the — quantities ; subtract the lat¬ 
ter from the former. 


Find the value of 


1 - i+i— tV 

2. I—|+f. 

3. f—|+i- 


4. 1—*• 

5. f— f f • 

6 . 14—S—Ar- 


7. TF-f+TF- 

8- f H~tV- 

9. 


10. 2|-ff—f—If. 

11. 5f—3|+2i—J. 

12. Tf—2f—8f—f. 

13. 18|—3J+ If—4f. 

14. 25f— 17 t 3 o + 16^— 6if. 


15. 14f+8|——7if. 

16. 4f—7f-j-8 x 7 2—4ff. 

17. 21 t 3 q—I 82 V—4JJ+14 x 9 ^. 

18. 17*—25H+6f+lf. 

19. 14f-5y-182T+16f . 


20 . Write the smallest and greatest fractions possible with 
the figures 3 and 7, 4 and 9, 6 and 11. Find the sum and 
difference of each pair. 

21 . Find the sum and the difference of the smallest and 
greatest of the fractions f, ff, and 

22 . A cubic foot of gold weighs 1203f pounds ; the same 
amount of silver weighs 625f pounds. Find the sum and 
difference of these numbers. 

23. A barrel of flour weighs 196 pounds. From a barrel 
of flour a grocer sells 24f pounds, 30f pounds, and 49J 
pounds. How much is left in the barrel ? 





FRACTIONS. 


21 


A 27J B 31# c 21# D E 

!_l_L_!_] 

From A to E is 100 miles. How far is it from d to e ? 

25. A pupil has 5 hours in school. Recitations last f of 
an hour each for arithmetic and grammar, § of an hour each 
for history and geography, and an hour each for writing 
and spelling. How much time is left for study ? 

26. A lady in shopping, paid $1# for gloves, $3f for shoes, 
and $18^ for dress material. What change should she re¬ 
ceive out of two ten-dollar bills ? 

27. Simplify 3#—'7f—3f+9i+4£— 1*. 


133. (For reference.) The value of a fraction depends on 
the relative magnitude of its terms (Art.117). If both terms 
of a fraction are multiplied, or both divided, by the same 
number, the value of the fraction is not altered (Art.115). 
But if only one of the terms is multiplied or divided, the 
value of the fraction is altered. 

a. The numerator shows how many parts are taken. When 
the number of parts taken is increased, the value of the 
fraction is increased. 

b. The denominator shows into how many parts the whole is 
divided. When the number of parts is increased, each part 
is made smaller and the value of the fraction is diminished. 

I. A fraction is multiplied either by multiplying its nu¬ 
merator, or by dividing its denominator (Art. 134). 

4 X 3=-V 2 -, or fx3=|. 

II. A fraction is divided either by dividing its numerator, 
or by multiplying its denominator (Art. 138). 

H-8 =h or #-*-8—ft. 





22 


THE 20th CENTURY ARITHMETIC. 


134. MULTIPLICATION OF FRACTIONS. 


The Product of an Integer and Fraction. 


I. 


II. 



5 times 3 boys are 15 boys. Directions.— To find the 

5 times 8 fourths are 15 fourths. P^uct of an integer and frac- 
p. 3 15 tion, multiply the numerator 

^ 4 4 of the fraction by the integer 

and place the product over the denominator. Cancel where possible. 


3 


Find the value of 


1 . 

8 X£. 

11. 

14Xy. 

2. 

2X*. 

12. 

17.X f • 

3. 

4Xf. 

13. 

9X|. 

4. 

8 Xf. 

14. 

bx t V 

5. 

6 X^. 

15. 

24X«. 

6. 

7XjV 

16. 

3X 2 t- 

7. 

5X t 3 t- 

17. 

4X t 2 t. 

8. 

8 X A- 

18. 

7X|. 

9. 

8 Xf. 

19. 

8 X 2 X 4- 

10. 

4XJ. 

20. 

11 X -£s 


21. 

1X9. 

31. 

A X 48. 

22. 

iV(j X4- 

32. 

1X89. 

23. 

AVX17. 

33. 

jX84. 

24. 

fVX 8 . 

34. 

HX 86 . 

25. 

¥X15. 

35. 

AX 49. 

26. 

i?X 14. 

36. 

AX 56. 

27. 

fiX 88 . 

37. 

HX84. 

28. 

VAX 80. 

38. 

HX42. 

29. 

2 V 3 X 2 I. 

39. 

MX 72. 

30. 

11X49. 

40. 

1X52. 















FRACTIONS. 


28 


135. The Product of Two or More Fractions. 


A 

1 1 


2 



3 

4 


s 


B 

1 



1 

2 


_ 

. 3 

i- 




Ci 

to 

CO 

1 

i 4 

1 3 

1 6 1 

7 

1 8 1 


— -1- 





1 

1 

1 1 


1-1 

-1-1- 

9 10 11 

I-1- 

12 13 

1-1- 

1 4 

1 5 


B is | of A. C is § of B. Therefore, C=f of f of A. 

From this it may be seen that § of f of a line is A of it. 
|X|=t 8 5- Multiplying by f is the same as multiplying by 
2 and dividing by 8. 

The product of two or more fractions is equal to the product of 
the numerators divided by the product of the denominators. 
Cancel when possible. 

t 

5 f W = _5 
12 

4 3 


Examples 32 to 42 are the same as 5 to 15, Art. 122. 


1. 

fXf. 

16 . txfx«. 

31. 

f |X|f Xf f . 

2. 

!Xf. 

17. fX$Xf. 

32. 

AXAXff. 

3. 

tx j. 

18. IXfXf. 

33. 

f fXffXif • 

4. 

*x*. 

19- tx V-x A- 

34. 

If XfyX A- 

5. 

IX f. 

so. AXfIXH- 

35. 

HXHXH* 

6. 

fxf. 

21. IfXlfXl. 

36. 

2 5 Y 3 4 Y 18 

5 I A- 4 5 xx 3 0 * 

7. 

*x*. 

22 . mx-v-xa* 

37. 

HXifXV 3 -. 

8. 

Ax*. 

23. ttXHXH- 

38. 

3 0 Y 2 8 Y 4 5 
45 A 44 AT05- 

9. 

f X f. 

24. fX |X If • 

39. 

45 Y 5 2 5 Y 3 6 
^75AT05A4? 

10. 

IX*. 

25. itXxXtt. 

40. 

180 Y26 V73 
TFAHA^. 

11. 

AX*. 

26. AX AX If. 

41. 

4 3 Y 8 2 Y 5 2 5 
KAg^A 9T5-* 

12. 

A X A* 

27. A 4 5 X -Vs 0 x A* 

42. 

wxmxii. 

13. 

ttX§. 

28. IfXTVxXfff. 

43. 

IXfXAXV 3 - 

14. 

AX 2 4 T- 

29. IXHXtt- 

44. 

tA Xff Xff. 

15. 

2tX|- 

30. y| X || X y| • 

45. 

«0)4. 

X 

“h 

X 

X* 

“4 







24 


THE 20th CENTURY ARITHMETIC. 


136. 

The Product of a Mixed Number and an Integer. 


24 

88 

5 | 72= 14f 
192 


206| 


1. 27X4|. 

2 . 35 x 8|. 

3. 17x9f. 

4. 43x25* 

5. 23 Xl8*. 

6. 65 x 82|. 

7. 7 X65*. 

8. 19x82* 


Multiply 24 by 8| (or 8f by 24). 


Directions. —Multiply 
the integer by the num¬ 
erator of the fraction ; 
divide by the denomi¬ 
nator as shown. Then 
multiply together the 
integers. 


88 

24 

5 | 72= 14f 
32 
16 

206f 


Examples. 


9. 

27X39J. 

17. 

10. 

8 X101f. 

18. 

11. 

21|X14. 

19. 

12. 

31|x6. 

20. 

13, 

18fx15. 

21. 

14. 

8|x 17. 

22. 

15. 

13*x25. 

23. 

16. 

38fX 23. 

24. 


45^x37. 25. 

72f x45. 26 . 

34|x 81. 27 . 

75f x 10. 28 . 

100* x 325. 29 . 
285x821*. 30 . 
410HX463. 31. 
347x642*. 32 . 


518* X 530. 
175x350*. 
643* x 700. 
741X641*. 
325if X800. 
720x892*f. 
365* X 255. 
314X513** 


137. The Product of Two Mixed Numbers. 


3 | 490 
4 


Multiply 245| by 38§. 


245f 
38| 

Directions. —Multiply first by 
the fraction, then by the integer. 

Copy this example, and notice 
whence everything is obtained. 


9502* 


*=4 

— 163*=| X 245 

114=28*=38 X | 

1960 l QQ 
735 i =38X245 







FRACTIONS. 25 

Examples. 


1 . 

751X 8 J. 

11. 34$X18$. 

21. 

4831X251. 

2. 

271X3$. 

12. 251X16*. 

22. 

255-frXlO!. 

3. 

18iX6f. 

13. 181X251- 

23. 

600* X 63y 7 5 . 

4. 

251x7*. 

14. 271X43$. 

24. 

194$ XlOf. 

5. 41fX8£. 

15. 75iX48f. 

25. 

236* X 42*. 

6. 

83* X9§. 

16. 83*X49$. 

26. 

400|X 17$f. 

7. 

54|x8f. 

17. 521X281. 

27. 

137*X25*. 

8. 

97lX5f. 

18. 25|X27|. 

28. 

314*x33f. 

9. 

87$ X 6 $. 

19. 371X951- 

29. 

149* X 44$. 

10. 

25$ x7$. 

20. 87|X26$. 

30. 

177* X 55*. 


138. DIVISION OF FRACTIONS. 


(See Art. 188.) 



a. 34 is contained in 1 two times. c. 34 is contained in 1 four times. 

b. K is contained in 1 three times, d. % is contained in 2 eight times. 


Flow many times is 

1 . | contained in 2 ? 

2 . contained in 2 ? 

3 . i contained in 2 ? 

4 . } contained in 1 ? 

5 . ^ contained in 1 ? 


In 

8 ? 

In 

5? 

In 

8 ? 

In 

4? 

In 

6 ? 

In 

7? 

In 

8 ? 

In 

4? 

In 

7? 

In 

2 ? 

In 

5? 

In 

6 ? 

In 

2 ? 

In 

4? 

In 

5? 










.26 THE 20th CENTURY ARITHMETIC. 




e. % is contained in 4 six times. 

Analysis. —Since X is contained in 4 twelve times, % is contained 
in 4 half as many, six, times. 



f. % is contained in 3 four 
times. 

Analysis. —Since 34 is con¬ 
tained in 3 twelve times, % is 
contained in 3 one third as 
many, four, times. 


a. 

b. 

c. 

d. 

e. 

f. 


1 divided by 4=2. 
1 divided by 4=8. 

1 divided by 4=4. 

2 divided by 4=8. 
4 divided by |=6. 


Notice that 1X4=2. 
Notice that 1X4=3. 
Notice that lXy=4. 
Notice that 2Xy=8. 
Notice that 4X|=6. 
Notice that 8X|=4. 


8 divided by |=4. 

In these examples we see that inverting the divisor and 
multiplying obtains the correct quotient. 


The Quotient of a Fraction and an Integer. 

139. How many times is f contained in 4 ? 

Analysis. —One fifth is contained in 4 twenty times; two fifths is 
contained half as many, 10, times. 

*4--f=4xf=10. Ans. 







FRACTIONS. 


27 


How many times is 

1. § contained in 4 ? 

2. f contained in 6 ? 

3. f contained in 9 ? 

4. f contained in 6 ? 

5. f contained in 8 ? 


6. | contained in 10 ? 

7. f contained in 6 ? 

8. | contained in 15 ? 

9. f contained in 12 ? 
10 . J contained in 14 ? 


Perform the divisions 




11. 8h-|. 

16. 14- 


21. 35-5-#. 

26. 56-5- 

12 . 15-5-#. 

17. 20- 


22. 40-5-#. 

27. 63-5- 

13. 18-5-#. 

18. 25- 

-*• 

23. 45-5-#. 

28. 28-5- 

14. 21-5-#. 

19. 28- 

_4 

7 . 

24. 22-5-#. 

29. 24-5- 

15 . 36-5-#. 

20. 21- 

-1- 

25. 18-5- x 6 x . 

30. 81-5- 


140. What is # of V 5 - ? (See Art. 188, II.) 


|XV-=f. Arcs. 


W/iai is 


iof V ? 
i of V ? 
i of Y 1 
iof |? 
i of if ? 


6 . 

7. 

8 . 
9. 

10 . 


\ of V ? 
i of Y ? 
i of if ? 

i Of Y ? 
i Of Y ? 


V--5-S=f. Arts. 

Perform the divisions: 

11. Y-^-3. 16. Y^O- 

12 . V-^-6- 17. if-5-2. 

13. y-5-3. is. y-^- 7 . 

14. y-s-n. 19. 

15. V--J-7. 20. H-^-9. 


141. The reciprocal of a number, whether integral or 
fractional, is 1 divided by that number. 

1 divided by is 3, and 1 divided by % is half of 3=4 ; there- 

1 __ 3 

fore the reciprocal of a fraction is the fraction inverted. 7 ^• 

142. To Divide a Fraction by a Fraction. 

Divide j by §. Analysis.— i is contained in 3 

15 times, and in times. § 

4 _f_ f == f Ans. is contained in f just half as 

many times; •£ of 1 ? 5 -=V"- A?1S. 










28 


THE 20th CENTURY ARITHMETIC. 


It may be noticed that \ 5 - is equal to f multiplied by 
(f inverted). Hence, for convenience, 

To divide a fraction by a fraction, invert the divisor and 
multiply. 

An integer is equivalent to a fraction with 1 for the de¬ 
nominator. 5=|. Cancel when possible. 


Examples. 


1. 

4- 


12. 


23. 

5 _jl_ 5 

^ • 2* 

CO 

lit" 

-HI- 

2. 

*- 


13. 

if-S-fJ. 

24. 

W- 

35. 

Mi¬ 

• 13 0 

• 4 4 T * 

3. 

i- 

-h 

14. 

4IJ-H' 

25. 

H-t- 

36. 

lt- 

.63 

4 0 * 

4. 



15. 


26. 

5_l_3 

^ * 7- 

37. 

12_l 

35 * 

-«• 

5. 

i~ 

2 

16. 

H-8. 

27. 

«+*■ 

38. 

60 _ 
T45 

— T G 0 6 5 • 

6. 

4- 

-2. 

17. 

7-|. 

28. 

8-s-i. 

39. 

3 3 0 
5T 

^“tV 2 5 • 

7. 

f~ 

-8. 

18. 


29. 

tt^tV 

40. 

1 3 2 _ 

2 3 1 

-ft* 

8. 

3 0 
4,9 

-Hf- 

19. 

J-i-5. 

30. 

6 -10 
^5 * 27* 

41. 

-w- 

-t 2 *V 

9. 

hi 

-ft. 

20. 


31. 


42. 

44- 

-w-. 

10. 

9- 

^3 

• 4* 

21. 

f-f. 

32. 

W-Afir* 

43. 

tWt 

-W-‘ 

11. 

n 

♦ 4 5 

. X 4- 

22. 

3 _i_3 

T • S' 

33. 

ff ■^"T4T* 

44. 

44- 

'/oV 

143 


Division of 

Mixed Numbers. 





Divide 186f by 4. 

Analysis. —Dividing as with integers, 
46^|. Ans a remainder, 2, is obtained. Reducing 

4 | 186y to an improper fraction and divid¬ 

ing by 4, we obtain If. 

a. When the divisor is a mixed number, it must be re¬ 
duced to an improper fraction ; and when both divisor and 
dividend are mixed numbers, both must be reduced. 






FRACTIONS. 


29 


Examples. 


1 . 125 ^ 4 . 

2. 321f-5. 

3. 422f-r-8. 

4. 718f-^6. 

5. 645f^-5. 

6 . 884|-=-6. 

7 . 291^-8. 

8 . 881|h-9. 


144. 


9. 498f-^4. 

10. 541 T 8 T -f-7. 

11. 745-5^. 

12. 1848-f-5|. 

13. 5684-^ 3f. 

14. 5280-^5^. 

15. 7842-=-4§. 

16. 1492-r-6f. 


17. 1776-f-5|. 

18. 1607-^8f. 

19. 1896-^5|. 

20. 3145-^9f. 

21 . 125 ^ 5 ^. 

22. 321f--8|. 

23. 5280f-^-5| 

24. 7842f-r-4f, 


25. 874-^-151. 

26. 884|-f-6f. 

27. 294^-5-J. 

28. 381§-^-8f. 

29. 498f^-5|. 

30. 541 T 8 T -^9f. 

31. 365^-f-7. 

32. 3142 4 g-;-5f. 


Miscellaneous Examples. 


1. I paid 12| cents a pound for a live turkey that weighed 
15f pounds. The dressed turkey weighed f as much as be¬ 
fore. How much per pound did the dressed turkey cost ? 

2. Add 5 to both terms of f and find the difference be¬ 
tween | and the resulting fraction. 

3. £ of a barrel of flour, containing originally 196 pounds, 
was spilled. If the barrel cost the grocer $5, at what price 
per pound must he sell the remainder to gain $1 ? 

4 . Subtract 5 from both terms of 4 and find the difference 
between the resulting fraction and 4 . 

5 . Three men are in business together. Mr. Muse owns § 
of it, Mr. Eads | of it, and Mr. Neel the remainder. The 
profits for one year were $5000. What was each one’s share 
of the gain ? 

6. Multiply both terms of f by 5 and find the difference 
between J and the resulting fraction. 

7. 52j gallons of water flow from one spring in 2 minutes ; 
183 gallons from another spring in 7 minutes. In five min¬ 
utes how much more flows from one than from the other ? 

8. | of a mile is 1980 feet. How many feet in a mile ? 








80 


THE 20th CENTURY ARITHMETIC. 


9 . There are 5| yards in a rod and 1760 yards in a mile. 
How many rods in a mile ? 

10. Two men together rent a pasture for one month, for 
$24. Mr. Howell puts in 5 horses and Mr. Candler 8 
horses. How much rent should each pay ? 

11 . There are 865 days in a year. What part of a year is 
a school session of 86 weeks, having 5 days to the week ? 

12. There are 5280 feet in one mile. Find the difference 
between f of a mile and J of a mile. 

13 . A bushel of oats weighs § the weight of a bushel of 
meal. A bushel of meal weighs f the weight of a bushel of 
corn. A bushel of corn weighs If the weight of a bushel of 
wheat. A bushel of wheat weighs 60 pounds. Find the 
weight of a bushel of corn, of meal, and of oats. 

14. A cubic foot of ice weighs 57§ pounds. How many 
cubic feet will weigh a ton, 2000 pounds ? 

15. Sound travels 4004 feet in 8f seconds. How far will 
it go in 6f seconds ? 


145. 


COMPLEX FRACTIONS. 


§=&=¥ (Art. 142.) 


3- Z 

Z Z 


a. *= 


b. -=i 

4 3 
Z T 


4X2 

Copy this example. Complete these examples. 

Perform the operations indicated and reduce to its simplest 
form : 

5 4 16 


1 . 4 - 


2 . 


7. 


12 . 


5| 

tt2 

8 


T 

8 81 
8 - fil¬ 


ls. 


3. 


4. 


9. 


6t 7 2 

9f 


10 . 


5. 

M 

Ti¬ 


ll. 


* tf 
6 . —. 
17 

sxf 


i—i 


14. 


8-f 

T 9 3+l 


15. 


2 

i 

8i+i' 






FRACTIONS. 


81 


16. 


20 . 


f °f T 4 7 

* 

f+i of 5 . 


4tt 

17 ‘ 6f+§i- 

» 1 92 

-3 


18. 


m+i 


19. 


i+i+f 


19§- 


2J 


21 . 


• 2# ' I X 2 - 22. (f+l + f)x(|—1 + f). 
i of i * - f * & 

23. (f— 2+!)-*-(i— }). 24. i—i- 


25. t—4x } 1 




i+i i+i i + i 

27. 3jX^ T + 6 ^~ 5| _gi. 

6i + 5f 5f 


26. 


29 


I. Zf_gi+3*? of 


S' 

i-(2|4-7|x|) 

82 

f+5f of f . 
I of 5f+f 


28. 


IfV 


30 H of 0 £ TT + I . 

* 2*—l*of (2i+l|) f- T \ 


146. REVIEW, 

l. Add ?i, f, H, and i. 2. Add f, f, f, J, and -ft. 

3. From f subtract A. 4. From T 7 2 subtract j||. 

•Efnd mhte 0 / 

5- Hf+3 T V—4f—G*+9f—2J. 6. 19Xf. 

7 . 3|XfXHX{|. 8. 26x825^. 9 . 85f+86 T 9 T . 

10. 28-s-Jf. 11. t 8 t 5 5 -H. 12. 5280-4-81. 13. 68J-*-91}. 

14. On what does the value of a fraction depend ? 

15. Give two ways of multiplying a fraction by 8. 

16. Give two ways of dividing a fraction by 8. 

17. A has 2f dollars, B has If dollars more than A, and 
C has as much as A and B together. How much have they 
all together ? 
















82 


THE 20th CENTURY ARITHMETIC. 


18. The sum of two fractions is 1 T 3 ^. One fraction equals 
f—What is the other ? 

19. The sum of the minuend and subtrahend is lyV ; the 
minuend is What is the subtrahend and what the dif¬ 
ference ? 

20. A pound of pure silver is worth 14f dollars ; of pure 
gold, 227f dollars. A pound of gold is worth how many 
pounds of silver ? 

21 . Name the results of addition, subtraction, multiplica¬ 
tion, and division. 

22. What kind of fractions can be added or subtracted ? 

23. A schoolboy is \ of an hour late every day for 86 
weeks (a school week is 5 days). If the regular session is 
5 J hours a day, how many hours is he in school, and how 
many does he miss ? 

24. Of what number is 25f both the divisor and quotient ? 

25. The sum of two fractions is 27|. If 3| be subtracted 
from their sum, the result will be 2 -J- more than the larger 
number. What is the smaller number ? 

26. If 13| be used as an addendum 23f times, how much 
less than 425J will the sum be ? 

27. A gold dollar weighs 25 t 8 q grains ; a silver dollar, 412^ 
grains. What will the two together weigh ? 

28. If the interest on $1 for 1 month is -J of a cent, and 
for 1 day, of a cent, how much is it for 5 months and 12 
days ? 

29. A day is ^£5 of a year ; a week, -fc ; and a month, y 1 ^. 
What part of a year is 1 day-f-1 week+1 month ? 

30. What part of a dollar is the sum of a cent of a 
dollar), a nickel ( ¥ V °f a dollar), a dime ( T V of a dollar), 
and a quarter (| of a dollar)? 

31. A gallon is of a cubic foot, a quart is a 
bushel is lyV How much is their sum ? 


FRACTIONS. 


33 


32. A cubic inch of water weighs 253 t 7 q grains ; of air, T % 
of a grain. What is the sum of their weights ? 

33. A cubic foot of gold weighs 1203f pounds, and of sil¬ 
ver, 625f. What is the sum of their weight's ? 

34. A dry quart measure contains 67f cubic inches ; a 
liquid quart, 57f cubic inches. How many more in the 
former than in the latter ? 

35. A silver dollar weighs 412| grains ; a gold dollar, 
25 t 8 o grains. What is the difference in weight ? 

36. The torrid zone is § of the earth’s surface ; the two 
temperate zones are if of the surface. What part of the 
surface is in each of the frigid zones ? 

a. Clocks at the following places would indicate the time set op¬ 
posite their names at the same instant (true sun time): 


New Orleans, 6. 

Washington, 

OH- 

New York, 7£. 

Havana, 

6-J. 

San Francisco, 3f. 

Philadelphia, 

7. 

Honolulu, 1 X V 

London, 

12. 


What is the difference in time between 

37. New York and San Francisco ? 

38. San Francisco and New Orleans ? 

39. Honolulu and San Francisco ? 

40. San Francisco and Havana ? 

41. San Francisco and Philadelphia ? 

42. London and Washington ? 

43. A cubic foot of water weighs 62^ pounds. What is the 
weight of 12§ cubic feet ? 

44. A cubic foot of ice weighs 58| pounds. What is the 
weight of 12f cubic feet ? 

45. If 12-fa silver dollars weigh a pound, how many gold 

dollars does it take ? 16 gold dollars weigh as much as 1 silver 

dollar. 

46. If a man smokes 5 cigars a day and the cigars cost 3 
for 10 cents, what will his cigars cost a year (365 days)?- 


84 


THE 20tli CENTURY ARITHMETIC. 


47. How much if he smokes 4 cigars a day at 5 cents each ? 

48. If it costs twenty-five cents for the first ten words 
and one and a half cents for each additional word, how 
much will it cost to telegraph this problem from New York 
to Washington ? 

49. A silver dollar weighs 412^ grains. If a gold dollar 
weighs 25§f grains, how many gold dollars weigh as much 
as one silver dollar ? 

50. At- § of a mile a minute, how long would it take a 
train to go from New York to Boston, 280 miles ? 

51. If 4J bushels of wheat make a barrel of flour, how 
many barrels of flour will 891f bushels of wheat make ? 

b. Postage on letters to any part of the United States costs “two 
cents per ounce or fraction thereof and on books, u one cent for each 
two ounces or fraction thereof 

What is the postage on 

Letters. Books. 

52. If ounces ? 2f ounces ? 55. ounces ? 7-J ounces ? 

53. § of an ounce ? 1J ounces ? 56. 4f ounces ? 8f ounces ? 

54. f of an ounce ? 8| ounces ? 57. 8J- ounces ? 2J ounces ? 

58. There are 8 pints in a gallon. A pint of water is 

added to a gallon of milk. What part of the mixture is milk ? 

59. Gunpowder is f saltpetre, j 1 ^ sulphur, and ^ char¬ 
coal. How many pounds of each in 1 ton (2000 pounds) of 
powder ? 

60. If a nickel weighs of a pound, what is a pound of 
nickels worth ? 

61. Water weighs 1000 ounces and milk 1080 ounces per 
cubic foot. What would a cubic foot of the mixture weigh 
which contains 7 parts of milk and 8 of water ? 

62. A cubic foot of silver weighs 625| pounds and is worth 
$18544. What would 100 pounds of silver be worth ? 


FRACTIONS. 


35 


63. A cubic foot of gold weighs 1203f pounds and is worth 
$398098. What would 100 pounds of gold be worth ? 

64. Around a circle is 3^- times the distance across it. 
W hat is the distance across a circle which is 5280 feet 
(1 mile) around ? 

65. The Bible contains 3566480 letters. 

i of the number of letters-f-471496=the number of words ; 

^4 of the number of words-f- 25598=the number of verses ; 

- 2 J f °f the number of verses -j- 259-=the number of chapters ; 

in? the number of chapters-}- l=the number of books. 

The word and occurs an average of 70l£ times in each 
book. How many times does and occur in the Bible ? 


147. Given the Value of a Fractional Part, 
to Find the Whole. 


1 . 


2 . 


3. 


i 


Here are 
are j of all 
£ ‘ have I ? 

Cf)Cf> 


9 matches, which 
I have. How many 

These 6 apples are 
f of all I have. How 
many have I ? 

In f of a foot there 
are 8 inches. How 
many inches in -J of a foot ? How many inches in 1 foot ? 

4. | of my height is 4 feet. How high am I ? 

5. | of the number of presidents to 1897 was 20. How 
many were there before 1897 ? 

If § of this dia- 


l A 


'A 


6 . 




X 


X 


gram is 21 inches, 


what is the whole length ? 
















86 


THE mil CENTURY ARITHMETIC. 


- 16 - 


7 . 


l /l 

Vi 

l A 

Vi 


what is the whole length ? 

^- 20 - 

8 . 


— --r-If f of this dia- 

/7 y? j ?'v ; gram is 16 inches, 


y>* 


a i 

Vs 



—-r-r—-j If | of this dia- 

[ V'& J H j gram is 20 inches, 


How 


what is the whole lengt 

9 . If ^ of my money is $88, how much have I ? 

10. If T \ of my money is $80, how much have I ? 

11. f of Washington Monument is 870 feet high, 
high is the monument ? 

12. The word and occurs 46278 times in the Bible. If 
this is T V of the whole number of words, how many words 
in the Bible ? 


Note. —There is no principle in arithmetic more important than the 
one involved in this Article. In order to develop it, the Committee 
of Ten recommend the introduction of the equation into arithmetic. 

Among several marbles, or toys, which a child has, it must often 
happen that one, which he wishes to identify, is so nearly like another 
that he cannot tell which is which. Thus originates the notion of 
equality. The things which we call equal—lines, weights, tempera¬ 
tures, sounds, colors—are things which produce in us sensations that 
can not be distinguished from each other. 

From familiarity with organic forms, there arises simultaneously 
the ideas of simple equality and equality of relations. Every process of 
arithmetic involves one or both of the ideas of equality and ratio: the 
idea of number itself involves them. A number is the ratio of a multi¬ 
plicity of objects to one of them, and the units composing the number 
must be absolutely equal , if the results of numerical operations are 
to be depended on. The only way in which we can establish a nu¬ 
merical relationship between units that are not alike, is to divide 
them into parts which yield units that are alike. 

The earliest mode of conveying the idea of a number of things, is 
by holding up as many fingers as there are things—using a symbol 
which is equal , in number, to the group symbolized. 

The highest function of the mind is that of comparing. The author 
has taken infinite pains to use the best methods for developing this 
function. 






















EQUATIONS. 


37 


EQUATIONS. 

148. An equation is an expression of equality. The 
first member is on the left of the sign of equality ; the sec¬ 
ond member is on the right of that sign. 

2x=Q is an equation in which 2x is the first member and 6 the 
second member, x stands for some unknown quantity. 

The following are self-evident truths called axioms: 

I. If two equal quantities be multiplied by the same number, 
the products are equal. 

If the value of a book, which we will call x, is $3, twice that value 
is $6. That is, if x=S, 2x=6. 

II. If two equal quantities be divided by the same number, the 
quotients are equal. 

If 2x=6, dividing both members by 2, we get ^=3. 


149 Find the value of 


1. 

5x when x= 

-4. 

6. 12a; when £=§. 

11. 

25a; when x— T \ . 

2. 

lx when x= 

-3. 

7. 7a; when x= T %. 

12. 

33a; when £= T 4 T . 

3. 

9£ when x- 

=7. 

8. 9a; when £=§. 

13. 

18a; when x=f^. 

4. 

8x when x= 

=4. 

9. 10a; when x—%. 

14. 

42x when £= T 5 ? . 

5. 

llx when x■ 

=5. 

10. 15a; when £=§. 

15. 

56a; when £= T 7 §. 

150. Find the value of x 

when 




1. 

5.e=15. 

11. 

4x=lS. 

21. 12a;= 

=92. 

31. 

fa;=8. 

2. 

7£=14. 

12. 

5x=-23. 

22. IGa — 

= 144. 

32. 

§£=15. 

3. 

3£=21. 

13. 

6x=27. 

23. 12a;- 

= 192. 

33. 

fa;=20. 

4. 

9£=36. 

14. 

3a;=20. 

24. I7a;= 

= 146. 

34. 

§£= 15. 

5. 

8z-=40. 

15. 

7x=24. 

25. 18£= 

=365. 

35. 

§£=12. 

6. 

4x=2S. 

16. 

9a;=30. 

26. 23a;= 

=521. 

36. 

fx=4. 

7. 

5a:—80. 

17. 

8x—35. 

27. 31a;- 

=400. 

37. 

§£=10. 

8. 

6.r=42. 

18. 

6x=40. 

28. 16a— 

=462. 

38. 

§£=6. 

9. 

3a;—83. 

19. 

4x=17. 

29. 15a;— 

=325. 

39. 

§£=15. 

10. 

9.c=99. 

20. 

9x=47. 

30. 27a;= 

426. 

40. 

££=14. 




88 


THE 20th CENTURY ARITHMETIC . 


151. It is very convenient to use the letter x to repre¬ 
sent the value, time, etc., of what is unknown. Thus : 

a. A merchant marks a suit of clothes to sell for $24. The 
trousers are marked twice as much as the vest, and the coat 
three times as much as the vest. What are the separate 
pieces marked ? 

Let #=the price of the vest; 
then 2x—the price of the trousers, 
and 3x= the price of the coat. 

6.r=the price of the whole suit=$24. 

6x=$24, price of suit. 2.r=$8, price of trousers. 

£=$4, price of vest. 8. / c=$12, price of coat. 

Problems. 

1. A father and son together weigh 240 pounds. The 
father weighs 8 times as much as his son. What does each 
weigh ? Let #=the weight of the son. 

2. A house and lot cost $4,500. The house cost twice as 
much as the lot. What was the cost of each ? Let a:=the 
cost of the lot. 

3. Mr. Man made a journey of 894 miles. He traveled 
twice as far by rail as by boat. How far did he go by rail ? 
Let x— the distance he traveled by boat. 

4. A tree growing out of water has 8 times as much above 
the surface as below it. The total height of the tree is 92 
feet. What is the height above water ? Let #=equal the 
number of feet below the surface. 

5. A bicyclist traveled 150 miles in 8 days. The first 
day he went 8 times and the second twice as far as on the 
last day. What was the distance traveled each day ? 

Let 4*=the distance he traveled the last day. 

6. Two numbers when added make 72. The larger is 
twice the smaller ; what are the numbers ? 


EQUATIONS. 


89 


7. •_ N g Y - » From Atlanta to 

Boston is 1090 miles. From Atlanta to New York is 4 times 
the distance from New York to Boston. What is the dis¬ 
tance from New York to both places ? 

8. An iceberg extends 9 times as far below the surface of 
the ocean as above it. If the total height of an iceberg is 
5280 feet, how high is it above water ? 

9. A bucket full of water weighs 15 pounds. If the 
water weighs twice as much as the bucket, what is the weight 
of each ? 

10. If a ball is dropped from a tower it will fall 64 feet in 
2 seconds. During the 2d second it falls 8 times as far as 
during the 1st second. How far does it fall in the 1st second ? 

11. A man bought an equal number of lamps, wicks, and 
chimneys for 8900 cents. Each lamp cost 800 cents ; each 
wick, 10 cents ; and each chimney, 15 cents. How many of 
each did he buy ? 

12. Two boys, 180 miles apart, are riding towards each 
other on bicycles ; one rides 7 and the other 8 miles an hour. 
How many hours before they will meet ? 

13. A bicyclist, 84 miles from home, is riding five times as 
fast as he could walk. 8 hours afterwards he breaks down, 
and after walking 2 hours, reaches home. How fast was he 
riding ? Draw a diagram. 

14 . A pile of dollar coins, some gold and some silver, 
weighs 816 ounces. A silver dollar weighs 16 times the 
weight of a gold dollar. How many dollars of each kind are 
there ? 

15. 60 pounds of wheat or 56 pounds of shelled corn make 
a bushel. In a car-load of wheat and corn, having an equal 
number of bushels of each, the grain weighs 89440 pounds. 
How many bushels of wheat (or corn) on the car ? 



40 


THE 20th CENTURY ARITHMETIC. 


Miscellaneous Examples. 

152 | of a number is 9, what is the number ? 

Analysis.— Let r=the number. Analysis.— (Without using x). 

f£—9 Statement. J of the number=9 

—^ of the number=3 
£=12 Ans. | of the number=12 Ans. 

Never express such a mathematical untruth as j=9. 


Examples. 

1. | of a number is 18, what is the number ? 

2. f of a number is 12, what is the number ? 

3. § of a number is 5, what is the number ? 

4. | of a number is 9, what is the number ? 

5. | of a number is 15, what is the number ? 


6. 15 is | of what number ? n. 

7. 8 is | of what number ? 12 . 

8 . 9 is | of what number ? 13 . 

9. 12 is f of what number ? 14 . 

10. 18 is f of what number ? 15 . 


f of what number is 12 ? 
f of what number is 18 ? 
I of what number is 21 ? 
f of what number is 85 ? 
f of what number is 42 ? 


16. i of the number of the States is 25 (1896). How many 
are (here ? 

17. § of the height of Washington monument is 888 feet. 
How high is it ? 

18. f of the number of feet in a mile is 8520 feet. How 
many feet in a mile ? 

19. | of the surface of the earth is 144 million square 
miles of water. What is the whole surface ? How much 
land is there ? 

20. In 1890, ^ 4 , or 8,272,000, of the school children of the 
United States attended school. How many school children 
were in the United States ? 

21. In | of a bushel there are 12 quarts. How many quarts 
in | of a bushel ? 



FRACTIONS. 


41 


22. | of the distance from Atlanta to Washington is 436 
miles. What is the whole distance ? 

23. f of the length of the Mississippi River is 2635 miles. 
What is the total length ? 

24. A boy spent § of his money, f of the remainder was 
2§ dollars. What had he at first ? 

25. A farmer planted as follows : \ of his farm in corn ; 
i of the rest in wheat; § of what was then left in potatoes ; 
f of the remainder in onions ; and the balance, 8 acres, in 
fruit trees. How many acres were in his farm ? Begin, f of 
the remainder=8 acres. 


153. } is f of what number ? 

Analysis.— 

Let £=the number. 

Statement. 

i*-i of W 

£=# Ans. 


Analysis. —(Without x.) 
J is IX} 
f is (| of |) Xf 
t is | of (|Xf) 

I is | of | Ans . 


Examples. 


1. § is f of what number ? 

2. f is f of what number ? 

3. f is | of what number ? 

4. | is f of what number ? 

5. | is | of what number ? 


6. | is f of what number ? 

7. i*r is I °f what number ? 

8- I i s fr °f what number ? 

9. f is f of what number ? 

10. | is § of what number ? 


154. a. f of a yard of cloth costs f of a dollar. What 


does one yard cost ? 
Analysis.— 

Let #=the cost of one yard 
§£=$! Statement. 
i c=== $To 

Ans. 


Analysis. —(AVithout x.) 
f the cost of one yard— $f 
73 the cost of one yard=$j 1 2 3 4 5 o 
The cost of one yard=$TS Ans. 






42 


THE 20th CENTURY ARITHMETIC. 


Examples. 

1. | of a yard of cloth costs $6, what does 1 yard cost ? 

2. | of a yard of velvet costs $5, what does 1 yard cost ? 

3. | of a yard of silk costs $5, what do 2 yards cost ? 

4. f of a yard costs $5|, what do 8 yards cost ? 

5. f of a pound of tea costs $f, what do 5 pounds cost? 

6. | of a pound of tea costs $j, what do 7| pounds cost ? 

7. A newsboy has 25 papers. He sells f of them for 45 
cents. How much is that apiece ? 

8. An engine goes § of a mile in £ of a minute. How 
long will it take to go 1 mile ? 

9. A bicyclist rides f of a mile in f of a minute. How 
long willjhe take to ride 5 miles ? 

10. Sound travels T 2 T of a mile in f of a second. How long 
will it take to travel 1 mile ? 

b. § of a yard of cloth costs . How many yards could 
be bought for $7 ? Find, first, how much $1 will buy. 

Analysis. —buys f of a yard. Statement. 
buys f of a yard. 

$1 buys -V°- of a yard. 

$7 buys V 0 ' of a yard. 

$7 buys 7f yards. Ans. 

11. $5 buys 2^ yards of cloth. How many yards will $8 
buy? 

12. 8 cents buys 5 lemons. How many lemons could be 
bought with 12 cents ? 

13. 8 pounds of coffee cost $|. How many pounds could be 
bought for $5 ? 

14. 2 pounds of candy cost . How many pounds could 
be bought for $8 ? 

15. | of a pound of candy costs How many pounds 
could be bought for $5| ? 


EQUATIONS. 


48 


16. f of a bushel of apples costs f of a dollar. How many 
bushels could be bought for $11| ? 

17. A train from Baltimore to Washington has gone f of 
the distance and has still 15 miles to go. What is the dis¬ 
tance between the two cities ? 

18. Two trains, one from Chicago the other from New 
York, travel towards each other. One has gone and the 
other of the whole distance, and they are still 555 miles 
apart. What is the distance from Chicago to New York ? 

19. | of a ton of coal costs $8-^. How many tons can be 
be bought for $15| ? 

20. Capt. Lowry owns f of the shares of a bank and Mr. 
Inman the remainder, f of the difference between the num¬ 
ber of their shares is 80. How many shares are there ? 


155. Examples in Fractions. 

1. I lost | of my money, gave away £ of the remainder, 
and had $10 left. How much did I have at first ? 

2. Arrange X 9 T , |f, andff in their order of magnitude. 

3. From what number can 8J be taken 8 times and leave 
no remainder ? 

4. If the product of two fractions is and one of them 
is |, what is the other ? 

5. What number exceeds 7f by 2^ ? 

6. The divisor is 1^, the quotient 2f, and the remainder 
J. What is the dividend ? 

7. To what fraction must ^ be added to give f ? 

8. If apples are bought at the rate of 5 for 2 cents and 
sold at the rate of 4 for 8 cents, how many must be bought 
and sold to make a profit of $4^ ? 

9. f of 42 is | of what number ? 

10. f of 82 is | of how many times f of 14 ? 



44 


THE 20th CENTURY ARITHMETIC. 


11. ^ of a certain number is 2 more than f of 20. What is 
the number ? 

12. | of a certain number is 5 less than f of 49. What is 
the number ? 

13. | of 28 is 5 more than § of what number ? 

14. What number is that f of which exceeds § of it by If ? 

15. Multiply the sum of f and § by their difference. 

16. Divide the difference between T2 and § by their sum. 

17. A owns | of a bale of cotton and B the remainder. 

The difference in the value of their shares is $10. What is 
the value of the bale ? 

18. A receives $£ for every day he works and gives back 

$f for every day he does not work. In 50 days he receives 

$80. How many days did he work ? 

19. A has 5 hours leisure time. How far can he ride on a 
street-car which runs 10 miles an hour if he returns walking 
4 miles an hour ? 

20. A can row 6 miles an hour in still water. He requires 
1^ hours to row 6 miles up a river. How long will he require 
to row 6 miles down the same river ? 

21. The circumference of the front wheel of a bicycle is 
2£ feet; of the rear wheel 2f feet. How many more turns 
will the rear wheel make than the front one in going 1 mile ? 
(1 mile=5280 feet.) 

22. When the time past noon is f of the time to midnight, 
what time is it ? (Draw diagram.) 

23. 4 times 8f is how many times 5 ? 

24. 85 is f of how many times 6 ? 

25. | of 25 is how many times 7 ? 

26. If 5 is added to both terms of j, how much is its value 
diminished ? 

27. If 8 is added to both terms of {, how much is its value 
increased ? 


FRACTIONS. 


45 


28. A boy bought apples at the rate of 8 for 2 cents and 
sold them at the rate of 2 for 8 cents. How much did he 
make on 60 apples ? 

29. Two trains are approaching each other. Number 1 is 
800 yards long and going 1760 yards per minute ; number 2 
is 400 yards long and going 1000 yards per minute. How long 
will they be in passing ? (Draw diagram.) 

• 30. A horseman starts at 6 a. m. and rides 6 miles an hour. 
A bicyclist starts at 7: 45 a. m. and rides 9^ miles an hour. 
What will be the time when the bicyclist overtakes the 
horseman ? 

31. A man walks to a town, going at the rate of 8f miles 
per hour. He rests there Manhour and returns, riding at the 
rate of 7^ miles per hour. The whole time taken was 250 
minutes. How far was it to town ? 

32. In a mile race (1760 yards), A gives B 100 yards start 
and beats him by 20 yards. B ran the full mile in 6 min¬ 
utes. How long would it take A to run a mile ? (Draw 
diagram.) 

33. Take away § of a number. The difference between the 
number taken away and f of the remainder is 8. What is the 
original number ? 

34. Divide a distance 85 inches long so that one part will 

be f as long as the other. - ^ x nr . . —-—I—-- 

85 inches. 

35. f of a cubic foot of water weighs 428f ounces. What 
is the weight of one cubic foot ? 

36. A mile is divided into two parts. Part one is 2112 
feet long, which is § as long as part two. How many feet 
in a mile ? 

37. 8 men or 5 boys can do a piece of work in 8 days. 
How many men together with 4 boys can do the work in 4£ 
days ? 



46 


THE mil CENTURY ARITHMETIC. 


38. The owner of T 3 T of a large building sold T % of his share 
for $40500. What should he who owns | of the building 
get for | of his share ? 

39. A can walk 1 mile in 15 minutes and B can walk it in 
20 minutes. How many minutes start must A give B that 
they may come out even in a 5-mile trip ? (Diagram.) 

40. A boatman can row 6 miles an hour in still water. 
He rows for 4 hours down a stream which runs 3 miles an 
hour. How long will it take him to come back to his start¬ 
ing point ? 

41 . A and B jointly own a pasture, A 30 acres and B 18 
acres. A, B, and C each keep an equal number of cows in 
the pasture for one month. C pays $48. How should it be 
divided between A and B ? 

42. A city lot was bought and increased in value ^ each 
year for two years. At the end of the second year it Was 
sold for $5445. What did it cost ? 

43 . A man sold two houses for $2400 each. On one he 
gained £ of the cost price ; on the other he lost | of its cost 
price. How much did he gain or lose on the two houses ? 

44. An English sovereign contains 123 grains of standard 
gold, of which is pure gold. A dollar contains 23i£ 
grains of pure gold. A sovereign is equivalent to how many 
gold dollars ? 

45 . Diversion. —A and B went to market with 80 geese 
each. A sold his at 2 for $1, andB his at 3 for $1 ; to¬ 
gether they received $25. The next day A went to market 
alone with 60 geese ; he sold them at 5 for $2 and received 
only $24. Why did his 60 geese bring $1 less than the 60 
geese of both ? 


DECIMALS. 


47 


15 6. DECIMALS. 


Write one hundred. 100. 

What is - T \ of one hundred ? Write it 


What is of ten ? 
What is T \ of one ? 

Integers. 

rV of a thousand=100. 
to of a hundred = 10. 
tV of ten = 1. 


Write it. 

Try to write it. 
Decimals. 

T V of one= T V = .1 
tV of T V = *01 

tV of TFO = ToV^— *001 


157. bince decimals have the same scale, they may be 
expressed by the same system of notation as that used for 
integers. Decimals are, in fact, but the extension of the 
same system to numbers on the right of units. A period, 
called the separatrix, or decimal point, is placed to the 
right of units to mark the ending of integers and beginning 
of decimals. 


158. Units, tenths hundredths thousandths. 

How to read a decimal : Read the number as if it were 
an integer and add the name of the last order on the right. 
.5=five tenths. .63=sixty-three hundredths. 

•147=one hundred forty-seven thousandths. 

Read the following decimals : 

1 . .5, .63, .147. 4. .7, .51, .325. 7 . .2, .25, .631. 

2 . .3, .45, .268. 5 . .8, .32, .467. 8. .4, .78, .963. 

3. .9, .81, .645. e. .1, .75, .894. 9 . .6, .99, .999. 

Write the following as decimals : 

10 . 5 tenths. 63 hundredths. 147 thousandths. 

11. 7 tenths. 25 hundredths. 245 thousandths. 

12. 8 tenths. 76 hundredths. 391 thousandths. 

13. 2 tenths. 49 hundredths. 875 thousandths. 


48 


THE mh CENTURY ARITHMETIC , 


159. 8.14 is a mixed decimal. In reading a mixed 

decimal, the word and should .be used between the integer 
and the decimal. 


3.14=three and fourteen hundredths. 


Write: 


Read : 


4. 8 and 9 tenths. 

5. 75 and 6 tenths. 

6. 19 and 4 tenths. 


1. 26.4, 7.25, 8.141. 

2. 85.9, 8.16, 8.225. 

3. 48.6, 5.98, 5.682. 


NUMERATION TABLE. 


160 



OD 

a 

c3 



Name. oq ^ g fl ^ 
C3 g 2 <3 2 


Order. 7th 6th 6th 4th 3d 2d 1st 

Integral. 


1st 2d 3d 4th 5th 6th 

Decimal. 


Memorize the names to the right of units. 


161. To read a decimal : Name the orders from the 
right as if it were an integer and, from the nature of the no¬ 
tation, the name that falls to units place, with the termina¬ 
tion ths, is the name of the last order on the right. This 
method shows how to read it as an integer and gives the name 
of the last order on the right at the same time. 


Read 8.141592. 








DECIMALS. 


49 


Naming the orders from the right as if the entire number were an 
integer, the name millions falls to 3; therefore the right-hand order, 
2, is millionths. The number is then read: 


3 and 141 thousand 592 millionths. 
Read the number in the numeration table. 
Read the following decimals : 


1. 4.867. 

2. 3.141. 

3. 1.414. 

4. 3.281. 


5. 5.0077. 

6. 3.1416. 

7. 3.2808. 

8. 7.5738. 


9. 17.865. 

10. 5.2809. 

11. 365.24222. 

12. 1.4142136. 


13. 3.2808693. 

14. .477125. 

15. 73.0025. 

16. 62.32106. 


162. A decimal is a fraction whose denominator is ex¬ 
pressed by the principles of decimal notation. The denomi¬ 
nator is 1 with as many zeros annexed as there are decimal 
orders. 


163. If both terms of the fraction be multiplied by 
10, the value is not altered and the product is y 3 ^. .3=.30 
or .300 or .3000. Zeros on the right of a decimal do not af¬ 
fect its value. 

6.2100=6 and 21 hundredths. 

18.84600=18 and 846 thousandths. 


Read: 

1. 7.20, 14.610, 

2. 25.40, 43.9500, 


17.80, .850, 

50.100, .0800, 


30.10300. 

4.771200. 


164. Write: 

1. Five tenths. 8. 

2. Fifteen hundredths. 9. 

3. Seven tenths. 10. 

4. Six hundredths. li. 

5. Nine tenths. 12 . 

6. Twelve hundredths. 13. 

7. Eight hundredths. 14. 


One hundredth. 

Four tenths. 

Four hundredths. 
Twenty-five thousandths. 
Seventeen hundredths. 
Nine ten-thousandths. 
Five and seven tenths. 


50 


THE mh CENTURY ARITHMETIC. 


15. 8 and fourteen hundredths. 18. Eighty-one millionths. 

16. IT and one tenth. 19. 78 millionths. 

17. 48 and five thousandths. 20. 4 and 16 millionths. 

21. Eighteen and one hundred seventy-three thousandths. 

22. One hundred nine and eleven thousandths. 

23. Five and eighty-seven hundredths dollars. 

24. Forty-three and three hundred seven ten-thousandths. 

25. Eight hundred seven and eighty seven hundredths. 

26. 853 millionths. 31. 85 hundred-thousandths. 

27. 25 ten thousandths. 32. 784 ten-thousandths. 

28. 11 and 35 thousandths. 33. 63 hundred-thousandths. 

29. 3 and 14 ten-thousandths. 34. 85 and 7 hundredths. 

30. 42 and 42 hundredths. 35. 384 and 72 thousandths. 


ADDITION. 


165. Only abstract numbers of the same order, or con¬ 
crete numbers of the same kind, can be added or subtracted. 
Hence, to add decimals, place like orders under each other 
and add as with integers. Like orders will fall properly if 
the decimal points are placed under each other. 


h. 

k. 

m. 


n. 

a. .30103. 

d. 4.8667. 

r. 62.32106. 

X. 

365.2422. 

b. .47712. 

e. 1.06943. 

s. 39.3704. 

y. 

2204.6. 

c. .53251. 

Add: 

f. 3.14159. 

w. 15.4323. 

z. 

3937.04. 


1. Column h. 

2 . Column k. 

3. Column m. 

4. Column n. 

5 . a, d, and r. 


6. b, e, s, and y. 

7. c, f, w, and z. 

8. a, e, s, and z. 

9. c, e, r, and x. 
10. a, d, x, and y. 


11 . a, s, w, and z. 

12 . f, r, s, and w. 

13. d, e, s, and y. 

14. c, f, r, and s. 

15. e, f, s, and x- 



DECIMALS. 


51 


SUBTRACTION. 


166. Place like orders under each other and subtract as 
with integers. 


Explanation.— 4.867=4.86700 (Art. 163) and 
subtraction is performed as if the zeros 
annexed. 

(See table of decimals 


4.867 
8.14159 
1.72541 
From 

1. b take a. 

2. c take b. 

3. d take a. 

4. d take e? 


5. r take s. 

6. s take w. 

7. s take e. 

8. w take f. 


in Art. 165.) 

9. y take x. 

10. z take x. 

11. x take r. 

12. y take f. 


the 
were 

13. x take f. 

14. z take a. 

15. z take r. 

16. y take w. 


167. MULTIPLICATI ON. 

Decimals may be written and treated as common fractions. 

A. B. 

A written decimally is .8 
t 2 o% written decimally is .25 
tMo written decimally is .075 
T'allMo written decimally is .01875 
Multiply together any two fractions in column A : 

TO X ToVX TT>%U=TbU%T)- 

The denominator of the product contains as many zeros as 
are found in the denominators of both factors. Observe that 
the number of orders in the decimal fractions (B) is the 
same as the number of zeros in the denominator of the cor¬ 
responding fraction (A). Therefore, the product of tivo deci¬ 
mals contains as many decimal orders as are found in both 
factors . 

.8X .25=.075. .25 X .075= 01875. 

.6X.81=.186. .44 X .81 =.8564. 

Supply zeros in the product when necessary. 




52 


THE 20th CENTURY ARITHMETIC. 


Estimate results where practicable. For example, you 
should be able to say that the product of 7.15 and 3.029 is a 
little over 21. This is often sufficient to show where to place 
the separat'rix ; it is always a good check. 

168. Point off the following products : 


1 . 

Multiplicand. 

.25 

Multiplier. 

.3 

75 

2 . 

.31 

.6 

186 

3. 

.44 

.81 

3564 

4. 

.075 

.983 

73725 

5. 

3.14 

2.71 

85094 

6 . 

1.414 

1.609 

2275126 

7. 

.0875 

8.75 

765625 

8 . 

4.786 

12.7 

607822 

9. 

75.05 

62.5 

4690625 

10 . 

3.1416 

.0621 

19509336 

11 . 

58.67 

2.479 

14544293 

12 . 

4.3896 

.357 

15670872 

13. 

43.896 

.0735 

32263560 

14. 

2.951 

69.05 

20376655 

15. 

31.25 

.403 

1259375 

16. 

.4685 

3.007 

14087795 

17. 

7.126 

.83 

591458 

18. 

.6256 

9.9 

619344 

19. 

57.63 

.085 

489855 

20 . 

342.6 

.0073 

250098 

21 . 

1.894 

.365 

691310 

22 . 

3.621 

.708 

2563668 

23. 

.428 

.356 

152368 

24. 

34.567 

.325 

11234275 

25. 

2.1894 

2.16 

4729104 


DECIMALS. 


58 


Write the product: 




a. 

b. e. 

d. 

26. 

.8x .7 

.08x.7 8x .7 

.008x7 

27. 

• 8X.7 

•8X.07 .03 X .7 

.08 x .07 

28. 

.9 X .6 

.009 x .6 .9 x6 

.9x60 

29. 

.2x4 

20 X. 004 .02 x 40 

.008 X .04 

30. 

.8x .15 

.03X15 30 x.015 .003x150 

Multiply: 




a. 

b. 

c. 

31. 

6.21 by .467 

.621 by .467 

.0621 by .467 

32. 

8.14 by 2.71 

8.14 by .271 

3.14 by .0271 

33. 

1.414 by 1.609 14.14 by .1609 

1.414 by 160.9 

34. 

.888 by 1.098 

8.33 by 10.93 

.0833 by 1.093 

35. 

17.32 by 6.102 1.732 by 610.2 

1.732 by .06102 

169. 

To Multiply by 10, 100, 1000, 

Etc. 


A X 10 = 8. X 100 == 8. X 1000 = 8. 

The product of tenths by 10, hundredths by 100, thou¬ 
sandths by 1000, etc., is units. Therefore, when a decimal 
is multiplied by 10, 100, 1000, etc., the tenths, hundredths, 
thousandths, etc., of the multiplicand respectively, become 
units of the product. Where is the separatrix with refer¬ 
ence to units ? 

Multiply 8.14159 by one thousand. 

Directions. —Beginning at units place, name the orders succes¬ 
sively to the right. Place the separatrix on the right of the order 
whose product by 1000 is units: 8.14159X 1000=8141.59. 

Supply zeros when necessary: 1.41X1000=1410. 


170. Write the products : 

1. 5.468X100. 3. .47712X1000. 5. .8885X10. 

2 . 8.1415x1000. 4. 1.609X100. 6. 17.82x1000. 




54 


THE 20th CENTURY ARITHMETIC. 


7. 8.14x1000. 

8. 1.414x100. 

9. 8.65X100. 

10. 5.2X100. 

11. .47712x10000. 


12. .58251x1000. 

13. 8.1416x10000. 

14. 71.8X1000. 

15 2.5x1000. 

16. 61.4X100. 


17 . 2.4x10000. ' 

18. 1.609x10000, 

19. 4.31x1000. 

20 . .07x1000. 

21. 1.609x10000 


171. Contracted Multiplication. 


The most important digit in any multiplier is the one on 
the extreme left. A mistake in multiplying by it produces 
a greater error in the result than a similar mistake in mul¬ 
tiplying by any other digit. The digits decrease in impor¬ 
tance towards the right. Therefore, in decimals, we may 
reverse the multiplier and carry the work only to that 
degree of accuracy desired. 

Multiply 3.141 by 2.718. 


a. 3.141 
2.718 


25128 

3141 

21987 

6282 

8.537238 


b. 

The figures of the mul¬ 
tiplier are reversed in b in 
order that we may multiply 
by them in their order of 
importance. Use x in place 
of the separatrix. 


3.141 

817*2 


6282 


2198 

7 

31 

41 

_25 

128 

8.537 

238 


If we desire only 3 decimal places in the product, the work 
on the right of the vertical line in b may be omitted and we 
will then have contracted multiplication. 


172. When the order of the digits in the multiplier is 
reversed, the products of each digit of the multiplier by the 
digit in the multiplicand over it are all of the same order. 

2 times 1 thousandth=2 thousandths. 

7 tenths times 4 hundredths=28 thousandths, etc. 

The name of this product is the name of the order over 
units of the multiplier. 








DECIMALS. 


55 


173. Directions. —1. Reverse the order of digits of the multi¬ 
plier and put its units place under that decimal order to which you 
wish the product carried. 

2. Multiply by each digit of the multiplier, beginning with the 
digit of the multiplicand over it. Write the partial products with 
their right-hand figures in the same column. 

3. Point off as many places in the result as are indicated by the 
figure under which units of the multiplier is placed. 

For greater accuracy, multiply the digit to the right of the one 
over the multiplier and add its nearest tens to the first written 
product. 

If the product is desired correct to a given number of places, the 
work should be carried one further place, because from the nature 
of the multiplication, the right-hand figure of the product may be 
in error one or two units. 

174. Perform the following multiplications and notice 
how the multipliers are written : 

a. 3.1416x2.718; answer to 3 decimals. 

b. 3.14X.47712; answer to 3 decimals. 

C. 3.142x15.163; answer to 3 decimals. 

a. b. *c. 

3.1416 3.14 3.142 

817 x 2 21774, 361 x 51 

*Annex zeros to the multiplicand when necessary. 

Examples. 

1 . 3.141592 X 1.4142; product to 3 decimals. 

2. 2.718282 x 1.6093; product to 3 decimals. 

3. 1.609329 X .47712; product to 3 decimals. 

4. 1.732051x15.43235; correct to 3 decimals. 

5. .477121 X2.539977; correct to 3 decimals. 

6. 8.33888 X .0333333; product to 4 decimals. 

7. .0393708x39.37079; product to 4 decimals. 

8. 61.02705X .6102705; product to 4 decimals. 






56 


THE 20th CENTURY ARITHMETIC. 


9. 1.0935696X .04780103; correct to 5 decimals. 

10. .843434 x .0222222; correct to 5 decimals. 

The answers to the following examples should be correct to 
three decimal orders : 

11. A cubic foot of water weighs 62.3210608 pounds. 
What is the weight of .033333 of a cubic foot ? 

12. An English pound (£, a piece of gold money) is worth 
4.86667 dollars. What are 1.06943 pounds worth ? 

13. A gallon of water weighs 8.83888 pounds. What is the 
weight of .6931042 of a gallon ? 

14. One yard equals .9144 of a meter. What part of a 
meter is .4536 of a yard ? 

15. What is the weight of 2747.7167 cubic inches of water, 
if each cubic inch weighs .086126 of a pound ? 

16. Multiply 3.141592 by itself and obtain a product cor¬ 
rect to 5 decimal orders. 

17. Multiply 1.414213 by itself and obtain a product cor¬ 
rect to 5 decimal orders. 


175. DIVISION. 

There are three cases in division of decimals : 

M hen the number of decimal orders in the divisor is 
I. fewer than .6336^-. 176, 

II. equal to .021 -^.007, 

III. more than .75 -s-,003, 

the number of decimal orders in the dividend. 

What is the product of a divisor and quotient ? 

The number of decimal orders in the dividend is equal 
to the number in the divisor plus the number in the quotient. 

Hence, the number of decimal orders in the quotient is 
the number in the dividend minus the number in the divisor. 



DECIMALS. 


57 


176 . Case I. 

.85094-^.271=3.14. 

Direction. —Point off the quotient so that the product of it and the di¬ 
visor will equal the dividend. 

Point off the following quotients : 



Dividend. 

Divisor. 


1 . 

.85094 

.271 

314 

2. 

.12144 

.528 

23 

3. 

.6336 

.176 

36 

4. 

.10388 

.49 

212 

5. 

.8856 

.18 

492 

6. 

.17661 

.841 

21 

7. 

3.564 

.44 

81 

8. 

73.725 

7.5 

983 

9. 

85.094 

3.14 

271 

10. 

2.275126 

1.414 

1609 

11. 

76.5625 

87.5 

875 

12. 

6.07822 

4.786 

127 

13. 

469.0625 

75.05 

625 

14. 

19.509336 

3.1416 

621 

15. 

1.090652 

7.31 

1492 

16. 

393.715 

24.5 

1607 

17. 

.929280 

1.76 

5280 

18. 

17.97984 

4.32 

4162 

19. 

6367.704 

81.2 

7842 

20. 

61502.31 

8.31 

7401 

21. 

56.9348 

418 

1386 

22. 

2853.048 

.642 

4444 

23. 

1.603170 

2.35 

6822 

24. 

2360.346 

.681 

3466 

25. 

4109.12 

2.22 

1896 




58 


THE 20th CENTURY ARITHMETIC. 


177. 


Case II. 


Divide .6 by .8. totV— to X -*j°—§=2. Ans. 

The quotient of two decimals having the same number of deci¬ 
mal orders is the same as the quotient of the numbers considered 
as integers. 


Write the quotients 

of the following: 



l. .6 h-J 

3. 

11 . 

.048= 

.006. 

21 . 

.006-5-.006. 

2. .15-5- 

.05. 

12 . 

.056= 

.007. 

22 . 

.111-5-.003. 

3. 7.5-5- 

.5. 

13. 

.035-f- 

.005. 

23. 

.0012= .0012. 

4. .14-5- 

.02. 

14. 

.068= 

.009. 

24. 

.0242-5-.0121. 

5. 2.7-5- 

.3. 

15. 

.021= 

.007. 

25. 

.8428-5-.4214. 

6. .56-5- 

.07. 

16. 

6.33= 

.03. 

26. 

.9696=.3232. 

7. .21= 

.03. 

17. 

4.84-5- 

.04. 

27. 

.4812=. 1203. 

8 . 1.8-5- 

.6. 

18. 

6.06-f- 

.02. 

28. 

3.141=3.141. 

9 . .35-5- 

.07. 

19. 

.969-5- 

.003. 

29. 

1.414=1.414. 

10. .42-5- 

.06. 

20 . 

.612= 

.006. 

30. 

14.92=14.92. 

178. 



Case 

III. 




It is seen from Article 168 that zeros may be annexed to 
the right of a decimal without altering its value. 

6.21=.003=? 6.210-^.003=207. Ans. 

Annex a zero to the dividend and the problem comes under Case II. 

When the number of decimal places in the divisor exceeds the 
number in the dividend , annex zeros to the dividend till the 
number of decimal places is the same; the quotient may then be 
found by Case II. 


Perform the following divisions 


1. 18.84-r-.003. 

2. 1.5-*-.05. 

3. 7.5-5-.003. 

4. 1.05-*-.0005. 

5. 2.1-5-.0007. 

6. .75-;-.0003. 


7. 18.6-5-.0006. 

8. 35.64= 044. 

9. 787.26-f-.075. 

10. 850.94= .0271 

11. 187.5-5-.075. 

12. 35.64h-.0081. 


13. 34.43-f-.0011. 

14. 172.8-5-.012. 

15. 4.536-s-.0021. 

16. 1190.4-5-.031. 

17. 251.33-f-.041. 

18. 41676=92. 




DECIMALS. 


59 


179. To Divide by 10, 100, 1000, Etc. 

30-4-10=3. 300=100=3. 3000=1000=3. 

The quotient of tens by 10, hundreds by 100, thousands 
by 1000, etc., is units. Therefore, when a number is di¬ 
vided by 10, 100, 1000, etc., the tens, hundreds, thousands, 
etc., of the dividend respectively, become units of the quo¬ 
tient. Where is the separatrix with reference to units ? 

Divide 3141.59 by one thousand. 

Directions— Beginning at units place, name the orders succes¬ 
sively to the left; place the separatrix on the right of the order 
whose quotient by 1000 is units. 3141.59-4-1000=3.14159. 

Supply zeros when necessary : 3.61 = 100=.0361. 


Examples. 


1. 

54.63-4-10. 

li. 

314.-4-100. 

21. 

301.031-4-1000. 

2. 

314.15-s-lOO. 

12. 

.025-4-100. 

22. 

.9144-4-10. 

3. 

47.712-4-100. 

13. 

6.14-4-100. 

23. 

.4536-7-100. 

4. 

160.9-4-10. 

14. 

.024=100. 

24. 

39.37-4-100. 

5. 

833.5=1000. 

15. 

16.09-4-1000. 

25. 

15.43-4-1000. 

6. 

173.2-4-100. 

16. 

1.414-4-100. 

26. 

404.7=1000. 

7. 

31.4-4-10. 

17. 

o 

•i—i 

•i- 

GO 

H 

27. 

1543.=100. 

8. 

141.4-5-100. 

18. 

4.77-4-1000. 

28. 

283.75=100. 

9. 

5325.1=1000. 

19. 

271.4-10000. 

29. 

6214.-4-10000, 

10. 

71.8-4-100. 

20. 

87.54-10000. 

30. 

2200.=1000. 


180. To Divide by a Number Ending in Zeros. 

Divide 774.9 by 300. 

7*74.9 Using the mark x to indicate the changed position 
g qq of the separatrix, we have divided both dividend and 
f x divisor by 100. Has the quotient been altered? 
Notice the following : 

a. 2.583 b. 3.49 c. 3.1416 

9 X 0|31 X 4.1 


8 X 00| 7*74.9 


ll x 00]34 x 55.76 







60 


THE goth CENTURY ARITHMETIC . 


Examples. 


1. 624.8-^-40. 

2. 786.9-f-30 

3. 25.55-^70. 

4. 43.47-^-90. 

5. 74.62-=-50. 

6. 316.85-^600, 

7. 325.93-=-500 


8. 69.055-=-200. 

9. 4834.7-^-800. 

10. 250.04-^400. 

11. 8141.6-5-8000. 

12. 20.7-f-2800. 

13. 737.25-^7500. 

14. 102.01-5-10100. 


15. 834.84^-9000. 

16. 161.525-^-32500 

17. 39.375-7-7500. 

18. 62.5-^-5000. 

19. 42.08-^8000. 

20. 8141.59-5-7000. 

21. 365.241-=-6000. 


181. Contracted Division. 

Division, in decimals, is contracted by striking off a 
figure from the divisor at each successive division, instead 
of bringing down a figure from the dividend. Before begin¬ 
ning the division it is necessary to find out how many sig¬ 
nificant figures will be in the quotient, and retain one more 
than this number in the divisor. 

Divide 48.482147 by 3.141592, retaining three decimals in 
the quotient. 

Explanation. — Considering only 
the integers, 3 and 48, we see that 
there will be two figures in the inte¬ 
gral part of the quotient; these, with 
the three decimals required, make 
five figures, which will be in the 
quotient. Therefore, retain 6 sig¬ 
nificant figures in the divisor. 

In multiplying the divisor by the 
figures of the quotient, carry the 
tens from the product of the last 
figure cutoff to the first product put 
down. 


15.432+ 

3.14Wj2 [48.482147 
314159 
170662 
157080 
13582 
12566 
1016 
942 
^74 
63 







DECIMALS. 


61 


Retain 3 decimal orders in the quotients of the following : 


1. 48.482147-^3.141592. 

2. 51.162708-^37.54552. 

3. 33.072604-=-44.28125. 

4. 33.077291-89.63125. 

5 . 19875.75-;-138624. 

6. 458.96833-^7.8420679. 

7. 2007.40833-^-27.891208 


8. 1207.1818—-35.88333. 

9. 19256.7708-=-3579.431. 

10. 39.1393^-39.37079. 

11. 2170.88-=-277.274. 

12. 63.8402785-^-7.087009. 

13. .5327809224--.78539. 

14. 29.6988175-V-6.783648. 


182. Reducing a Common Fraction to a Decimal. 

Reduce ^ and f to decimals (to 4 decimal orders). 

.7143— .8571+ 

7 | 5. 7 | 6. 

When the next figure of the quotient would be 5 or more, 
increase the last digit by 1 and write — after it: .71428 is 
nearer .7143 than .7142; otherwise write +. The sign — 
shows that if something (not known) were subtracted from 
the quotient, it would be exactly right. 

Reduce to decimals, retaining four decimal orders: 



a 

b. 

e. 

d. 

e. 

1. 

3 

7* 

4 

9 • 

A- 

it- 

iif • 

2. 

tV 

T+ 

it- 

li¬ 

t¥t- 

3. 

T2- 

T3* 

li¬ 

ft- 

iff. 

4. 

f- 

H* 

lt- 

If 

tW+ 

5. 

9 

TT- 

*• 

it- 

«■ 

tYtV 

6. 

2+ 

8|. 

®A- 

6ft • 

9*. 

7. 

3 t + 

4 9 

7f. 

14ff- 

6441. 

8. 

6 t V 

8A- 

8 A- 

13 t 9 ^. 

37if. 

9. 

7tV 

9*. 

6f. 

19f^. 

91 t 4 A 

10. 

5 t V 

•5A- 


82§£. 

65 t 8 i% 






62 


THE 20tli CENTURY ARITHMETIC. 


183. Reducing a Decimal to a Common Fraction. 
Reduce .625 to a common fraction. 

.625 may be written to 2 o 5 o- tWV— Ans. 

In the same way any decimal may be reduced to a com¬ 
mon fraction. 

Reduce to a common fraction : 


1 . 

.625. 

8 . 

.1645. 

15 . .11025. 

22. .582. 

2. 

.1875. 

9 . 

.1155. 

16 . .275. 

23 . .882. 

3 . 

.375. 

10. 

.1925. 

17 . .1815. 

24 . .8085. 

4 . 

.425. 

11. 

.726. 

18 . .198. 

25 . .1115. 

5 . 

.675. 

12. 

.9075. 

19 . .126. 

26 . .15925. 

6 . 

.1225. 

13 . 

.5929. 

20. .735. 

27 . .1575. 

7 . 

.1525. 

14 . 

.13475. 

21. .935. 

28 . .1845. 


184. REVIEW. 

1. What is the scale of decimals ? 

2. What is the name of the point between integers and 
decimals ? 

3. What ought the subject “decimals” to teach ? 

4. Read 2.400421; 400.04. (Art. 161.) 

5. Write two hundred and four thousandths—204 thou¬ 
sandths. 

6. How many decimal places are in the product of two 
decimals ? 

7. What is the quotient of two decimals which have the 
same number of decimal places ? 

8. Write the largest decimal possible with the figures 
1, 4, 9, 2; with 1, 7, 7, 6. 

9. Arrange in their order of magnitude, fff, fff, and 
3.1415926. 

10. Add together the sum, difference, product, and two 
quotients of 3.14 and 1.414 (3 decimals). 



DECIMALS. 


63 


11- What decimal multiplied by 25 equals the sum of -§, 
A, I, .0885, and 8.1416? 

12. Simplify - 4 - X ^|^~|-^ X1 - 41 of $5. 

0.14—1.41 

13. Arrange in their order of magnitude, f-f-f, 
and 1.41421. 


14. Of what number is 8.1416 both the divisor and 
quotient ? 

15. Multiply 39.870482 by 3.1415926, contracted method, 
correct to three decimals. 

16. The exact solar year is .24222 of a day more than 365 
days. To how many days would this amount in 400 years ? 
If 97 extra days are added in 400 years, what would then be 
the error ? 

17. What is the easiest way of reducing A to a decimal ? 
In this way reduce the following to decimals : H, M, M, f. 


18. Reduce — to a decimal by multiplying both numera- 
tor and denominator by 12. Reduce —— to a decimal. 

m 

19. Reduce .34375 to a common fraction. 

20. Divide by the contracted method, retaining four deci¬ 
mal places, 15.4323487 by 3.1415927. 


185. APPLICATION OF DECIMALS. 

United States Money. 

The money used in the United States consists of metal 
coins made of gold, silver, and copper; and government and 
bank notes made of paper. 

Every country has its own money, which is subject to the 
laws in the country in which it is used. To Thomas Jeffer¬ 
son is due the simplicity of our money system. 




64 


THE 20th CENTURY ARITHMETIC. 


Table. 

10 Mills = 1 Cent (/) 

10 Cents = 1 Dime (d.) 

10 Dimes = 1 Dollar ($) 

10 Dollars =1 Eagle (E.). 

186. The scale of U. S. money is decimal; the decimal 
system of notation may, therefore, he used in writing it. 

$64,825 may he read, 6 eagles 4 dollars 8 dimes 2 cents 5 
mills. 

Practically, however, neither eagles nor dimes are named 
and the number is read, 64 dollars 82 cents 5 mills. It 
may also be read 64 and 825 thousandths dollars. This 
method is used whenever the number of decimal places 
extends beyond the 3d (mills). 

$5.2753 is read 5 and 2753 ten thousandths dollars. 


Read : 





1. $2.75. 5. 

$173.25. 

9. $872,252. 

13. 

345.2/. 

2. $35.20. 6. 

$275,875. 

10. $5280.275. 

14. 

285.5/. 

3. $4,825. 7. 

$6.2875. 

ll. $65.8765. 

15. 

50.4/. 

4. $25,862. 8. 

$75.3243. 

12. $125,625. 

16. 

365.5/. 

187. Write: 






1. Two dollars seventy-five cents. 

2. Thirty-five dollars fifteen cents five mills. 

3. Fifteen dollars twenty-three cents two mills. 

4. 25 dollars 6 cents 5 mills. 

5. 75 cents 3 mills. 

6. 13 and 643 thousandths dollars. 

7. 243 and 1846 ten thousandths dollars. 

8. 125 and 1492 ten thousandths dollars. 

9 . 642 and 2342 ten thousandths dollars. 

10. 849 and 325 thousandths dollars. 


U. S. MONEY. 


65 


Reduction. 

Eagles dollars. dimes cents mills. 

Memorize these names from left to right and from right to left. 

a. Reduce $3,875 to cents. 

Directions.— Beginning at units place, name the orders successively 

to the right. Place the separatrix on the right of the required de- 

nomination. $3,875=387.5/. 

b. Reduce 26413.5 mills to dollars. 

Directions.— Beginning at units place, name the orders successively 
to the left. Place the separatrix on the right of the required denomi¬ 
nation. 26413.5 mills=$26.4135. 

Reduce to cents : 

1. $3,875. 4. $7,645. 7. $35.61. 10. $75,891. 

2. $5,631. 5. $3,892. 8. $24,863. 11. $43,625. 

3. $9,425. 6. $2,485. 9. $45,326. 12. $13,845. 

Reduce to dollars: 

13. 26413 mills. 16. 75849/. 19. 35861.9/. 

14. 57846 mills. 17. 82765/. 20 . 83426.5/. 

15. 98745 mills. is. 95481/. 21. 7384.95/. 


189. Operations with U. S. Money. 


a. Add $25.65, $4,243, and $789.50. 

b. Subtract $31,243 from $789.50. 
C. Multiply $25.65 by 33. 

d. Divide $31,245 by 3. 


In business transac¬ 
tions, if the final result 
gives 5 or more mills, 
the cents is increased by 
1 ; if less than 5 mills, 
the mills are discarded. 


a. 

$25.65 

4.243 

789.50 


b. 

$789.50 

31.243 

$758,257 


c. d. 

$25.65 $10,415 

33 3 | $31,245 

7695 
7695 


$819,393 


$846.45 








66 


THE 20th CENTURY ARITHMETIC. 


Find the value of 


1. $47.55+$3.304-f$125+$.865. 

2. $55+$19.37£-}-$863.12+$.62£. 

3. $175-}-$7+$8-04+$.75+88/. 

4. $.96f X 45/ + $8f + $4.66£+95/. 

5. $95.021—$17,918. li. $185.18X4.25. 

6 . $61—$28,444. 12. $87iX48f. 


7. $18|—$4.8667. 

8. $285.16—$9|. 

9. $187X75. 

io. $5.80X1.6. 


13. $128-^8. 

14. $1.56^-12. 

15. $632^-$158. 

16. $41.25h-$125. 


17 . A merchant sold goods as follows : Monday, $215.18; 
Tuesday, $187.25 ; Wednesday, $97.65 ; Thursday, $105.47 ; 
Friday, $89.16; Saturday, $300.07. What was the total for 
the week ? What the average per day ? 

18. A had $150.75. He bought a bicycle for $105 and 
sold it for $130.50. He then collected an amount due him 
of $57.15, and paid two debts of $37.25 each. How much 
money did he then have ? 

19 . I pay $27.50 for a suit of clothes and hand over three 
ten-dollar bills. What change should I get ? 


e. In practice, change should be obtained by adding to the pur¬ 
chase money the amount required to make the money given in pay¬ 
ment. In the above, we would lay down a 50^ piece (or equivalent), 
saying $28 ; then $2, saying $30. 

What change should be returned when 


20. $10 is given to pay for a purchase of $8.35? 

21. $5 is given to pay for a purchase of $3.15 ? 

22. $20 is given to pay for a purchase of $14.85 ? 

23. $50 is given to pay for a purchase of $31.18 ? 

24. $10 is given to pay for a purchase of $3.46 ? 

25. $50 is given to pay for a purchase of $18.37 ? 


U. S. MONEY. 


67 


26. I bought 157 cattle in Texas at $16.85 a head. It 
cost $8.17 a head to get them to Chicago, where I sold them 
for $28.18 a head. What was my total gain ? 

27. John’s college expenses are $21.50 a month. How long 
will $204.25 last him ? 

28. From the sum of $125.60 and $65.08 take their differ¬ 
ence. 

29. Two men bought a city lot for $7845. They made 
improvements on it that cost $684.50, and then sold it at a 
loss of $218.25 to each. What was the selling price ? 

30. Three men bought a ship. A paid $5225, B. paid 
$800.50 more than A, and C paid $548.75 less than A and B 
together. What was paid for the ship ? 

31. A bought two farms at $7280 each, 25 shares of 
stock at $97| per share, and had $487.25 left. How much 
had he at first ? 

32. At $11.50 per thousand feet, how much lumber could 
be bought for $28.00 ? 

33. A man who owned of a store sold f of his share for 
$22000. What was the value of the store ? 

34. If the interest on $1 for one year is 6 cents, what is 
the interest for one month ? For one day ? (80 days=l 
month.) 

35. 6 dry quarts = 7 liquid quarts. If I buy 1020 dry 
quarts at 10 cents a quart and sell them as liquid quarts at 
10 cents per quart, how much do I gain ? 

36. What must I ask for a bicycle which cost $45 so that 
I may take $10 off the asking price and still make $12 
profit. 

37. What must I ask for a bicycle which cost $50 so that 
I may take off y 1 ^ of the asking price and still make a profit 
of f of the cost ? 


68 


THE mh CENTURY ARITHMETIC . 


f. If the interest on $1 for 1 year is 6 cents, for one month it is 
.i a cent, and for 1 day ^ of a mill, what would be the interest on $1 
for 3 years 9 months 12 days ? 

Interest on $1 for 3 years = 6X3 0 n cents) = $.18 

Interest on $1 for 9 months = £ of 9 (in cents) = .045 

Interest on $1 for 12 days = £ of 12 (in mills) = .002 

Interest on $1 for 3 years 9 months 12 days = $.227 

If the interest on $1 for one year is 6 cents , what would he the 
interest on $1 for 


38. 

Years. 

8, 

months. 

9, 

days. 

12? 

43. 

Years. 

4, 

months. 

5, 

days. 

15? 

39. 

2, 

7, 

18? 

44. 

8, 

9, 

21? 

40. 

4, 

11, 

24? 

45. 

2, 

11, 

9? 

41. 

8, 

8, 

18? 

46. 

8, 

7, 

27? 

42. 

2, 

5, 

6? 

47. 

5, 

9, 

21? 


48. There are 7000 grains in a pound avoirdupois. A gold 
dollar weighs 25.8 grains. How many pounds do 1000 gold 
dollars weigh ? 

49. 7000 grains=l pound. A silver dollar weighs 412.5 
grains. 1000 silver dollars weigh how many pounds ? 

50. A cubic inch of water weighs 252.458 grains. Iron 
weighs 7.23 times as much as water. 7000 grains=l pound. 
What decimal part of a pound (two decimal orders) does a 
cubic inch of iron weigh ? 


199. Aliquot Parts of 100. 


An aliquot part of a number is an exact divisor of it. 
Many operations of multiplication and division may be 
shortened by remembering the aliquot parts of 100. 


6i= T V of 100. 
25=i of 100. 
50=i of 100. 
75=f of 100. 


8£= T V of 100. 
16f=& of 100. 
33i=£ of 100. 
66|=f of 100. 


12£=-J of 100. 

87i=f of 100. 

62^=| of 100. 
S7i=i of 100. 




ALIQUOT PARTS. 


09 


191. Multiply 250X121. 

250X121=250X1 of 100=25000X1=8125. Am. 

What is the value of 

1.45x881? 6.144X81? n. 84x1.25? 16.7850x881? 

2. 64X 25? 7. 280X621? 12. 78X1.381? 17. 5280x121? 

3. 216X lOf? 8. 16x871? 13. 32x1.371? 18. 3.1416X.16J? 

4.124x75? 9.24X371? 14. 48X 1.121? 19. 365.24X .66|? 

5. 496X121? 10. 242x66f? 15. 36X1.16f? 20. 1492X.871? 


192. Divide 250 by 124. 

250-r- 121=250=i-li=250x T f, y =2.50x8=20. Am. 


What is the value of 

1. 350~-16f? 

2. 420-^-50 ? 

3. 75-^-334? 

4. 65—25 ? 

5. 128-f-75 ? 


6. 82-f-12| ? 

7. 84--37i? 

8. 63~-8£? 

9. 20-=-6^ ? 
10 . 93 ^ 62 ^? 


11. 42h-1.33J? 

12. 78-^1.25? 

13. 82^-1.16| ? 

14. 105-T-1.75 ? 

15. 348h-1.66§ ? 


193. @—at so much for each, per pound, etc. 

100/=$1, 50f=$i, 3325/=$±, etc. 
What is the cost of 250 pounds of beef @ 12|/ ? 

250xl2$/=250x$i=$31.25. Ans . 


i'Vad the cost of 

1. 42 knives @ 50/. 

2. 24 baskets @ 33£/. 

3. 132 books @ 75/. 

4. 82 pounds rice @ 6£/. 

5. 48 pounds cheese @ 16|/. 

6. 90 baskets @ 66f/. 

7. 45 bushels wheat @ 87|/. 

8. 37 pairs stockings @ 25/. 

9. 150 pounds beef @ 12^/. 

10. 30 handkerchiefs @ 37£/. 


11. 350 hats at 62^/. 

12. 75 pairs shoes @ 87-J-/. 

13. 128 books @ 75/. 

14. 233 clocks @ $1.33-£. 

15. 864 pictures @ $2.50. 

16. 296 balls @ 37£/. 

17. 343 rackets @ $4.25. 

18. 126 nets @ $3.75. 

19. 492 chairs @ $1.12J. 

20. 864 lamps @ 66|/. 


70 


THE 20th CENTURY ARITHMETIC . 


194. How many dolls @ 12£/ can be bought for $2.50 ? 
As many as 1234^ is contained in 250^. 

250-5-12£=250-f- J-^=250 X T ! o =2.50 X 8=20. Ans . 

How many 

1. Pounds of cheese @ 12-J/ can be purchased for $7.25 ? 

2. Pairs of gloves @ 50/ can be purchased for $7.50 ? 

3. Bushels of apples @ 33J/ can be purchased for $2.70 ? 

4. Bushels of wheat @ 66§/ can be purchased for $75.80 ? 

5. Knives @ 25/ can be purchased for $5.75 ? 

6. Handkerchiefs @ 37|/ can be purchased for $24 ? 

7. Dozen eggs @ 12£/ can be purchased for $6.50 ? 

8. Pounds of rice @ 8J/ can be purchased for $4.50 ? 

9. Books @ 75/ can be purchased for $12.75 ? 

10. Pounds of sugar @ 6£/ can be purchased for $2.45 ? 

195. When the price per hundred or per thousand is 
given to find the price of any number. 

At $1.65 per hundred, what would 828 oranges cost? 

Analysis. —How many hundreds in 328 ? 3.28. 
The question might, therefore, read : u If 1 hun¬ 
dred oranges cost $1.65, what will 3.28 hundreds 
cost ? ” Evidently 3.28 times $1.65. 

Directions. —Find how many hundreds are in 
the given number and multiply this by the price 
of 1 hundred. Similarly, if the price per thou¬ 
sand is given. 

C. stands for hundred; M. for thousand. 

What would 

1. 420 oranges cost @ $2.25 per C. ? 

2. 528 lemons cost @ $1.75 per C. ? 

3. 3280 bricks cost @ $4 per M. ? 

4. 11280 feet of lumber cost @ $15.50 per M. ? 

5. 235 bananas cost @ $1.20 per C. ? 


$ 1.65 
3.28 
1320 
330 
495 

$ 5.4120 





DECIMALS. 


71 


6. 125000 belgian blocks cost @ $6 per M. ? 

7. 120 miles of travel cost @ $25 per M. ? 

8. 840 melons cost @ $75 per C. ? 

9. 525 shingles cost @ $8.75 per M. ? 

10. 868 paving stones cost @ $19.50 per M. ? 

11. 8842 feet of lumber cost @ $17 per M. ? 

12. 9641 brick cost @ $11.25 per M. ? 

13. 7642 shingles cost @ $4.86 per M. ? 

14. 864 oranges cost @ $2.86 per C. ? 

15. 391 lemons cost @ $1.85 per C. ? 

16. A lumber dealer paid $8.30 per M. feet for plank and 
sold at $11.50 per M. What did he gain on 17 M. feet ? 

17. What will be the freight on 875 pounds at $1.60 
per C. ? 

18. It costs a railroad 63/ per C. to haul freight a certain 
distance. It charges 78/ perC. What is gained in hauling 
13864 pounds ? 

19. If it costs 13/ to put on a hundred palings, what will 
it cost to put on 3296 palings ? 

20. A train consists of 34 cars filled with cotton bales. 
The average number of bales per car is 37 and the average 
weight per bale is 428 pounds. How much will the road re¬ 
ceive, the freight rate being 67^/ per C.? 


196. When the price per ton is given, to find the cost of 
any number of pounds.' 

T. stands for ton ; lb. for pound. 

1 ton=2000 pounds. 

1. How many tons in 4000 pounds ? 

2. How many tons in 6000 pounds ? 

3. How are pounds reduced to tons ? Ans. Divide by 
2000. 



72 


THE goth CENTURY ARITHMETIC. 


197. Reduce 8642 pounds to tons. 

8642— 2000=8.642-^2=4.821. 
8642 pounds=4.821 tons. Ans. 

Reduce to tons: 


1 . 

4000 lb. 

6. 7250 1b. 

11 . 

185 lb. 

16. 

946 lb. 

2. 

5280 lb. 

7. 82 lb. 

12. 

28642 lb. 

17. 

784 lb. 

3. 

784 lb. 

8. 75 1b. 

13. 

688 lb. 

18. 

896 lb. 

4. 

1250 lb. 

9. 1492 lb. 

14. 

99 lb. 

19. 

675 lb. 

5. 

125 lb. 

10. 1776 1b. 

15. 

48684 lb. 

20. 

555 lb. 


198. At $5.25 per ton, what would 1280 pounds of ice 
cost ? 

Analysis. —1280 pounds=.64 of a ton. The 
question might, therefore, read: “If 1 ton 
of ice costs $5.25, what will .64 of a ton 
cost?” Evidently, .64 times $5.25. 


1. 4000 lb. of ice cost @ $6 per ton ? 

2. 5260 lb. of hay cost @ $12 per ton ? 

3. 780 lb. of coal cost @ $4.50 per ton ? 

4. 1250 lb. of salt cost @ $20 per ton ? 

5. 125 lb. of silver cost @ $24000 per ton ? 

6. 7250 lb. of steel rails cost @ $82 per ton ? 

7. 8892 lb. of ice cost @ $6.50 per ton ? 

8. 1492 lb. of hay cost @ $11.80 per ton ? 

9. 1776 lb. of coal cost @ $1.85 per ton ? 

10. 5 lb. of silver cost @ $24000 per ton ? 

11. 24860 lb. of iron cost @ $16.40 per ton ? 

12. 125 lb. of coal cost @ $5.50 per ton ? 

13. If it costs $1.85 per ton to manufacture ice and $1.28 
to deliver it, how much is gained in selling 1776 pounds @ 
$4 per ton ? 


$ 5.25 
.64 

2100 

8150 


$ 8.8600 
What ivould 




DECIMALS. 


73 


14. If it costs 85/ per ton to dig coal and 37£/ to trans¬ 
port it, at what per ton must it be sold to gain 50/ on 1000 
pounds ? 

15. If hay costs $7.60 per ton to market it, at what price 
per ton must it be sold to gain $1 on 500 pounds ? 


199. REVIEW. 

1. Express 16^/ as the decimal of a dollar. 

2. Multiply 417 by 38J, using quickest method. 

3. Divide 8172 by 12£, using quickest method. 

4. What would 525 oranges cost at $2.35 per C. ? 

5. What would 7245 bricks cost at $12.15 per M. ? 

6. If it costs $1.65 per ton to manufacture ice and $.85 
to deliver it, how much is gained in selling 3700 pounds at 
$5 per ton ? 

7. 27.7274 cubic inches of water weigh one pound, and ice 
weighs .93 as much as water. What is the weight of a cubic 
foot (1728 cubic inches) of ice? 

8. A train goes 275.25 miles in 4.75 hours. How many 
miles is that per hour ? 

9. From one spring 125.63 gallons of water flow in two 
minutes and from another 351.75 gallons in 7 minutes. 
How much more flows from one than from another in 5f 
minutes ? 

10. Two trains are 521.25 miles apart. They approach 
each other, one running at the rate of 42.3 and the other at 
37.5 miles per hour. How long before they will meet ? 

11 . I lent Will .25 of my money and Roy .8 of the re¬ 
mainder. Roy received $350 more than Will. How much 
money had I altogether ? 

12. From New York to Boston is 217 miles. How many rods 

is it ? 1 rod=.003125 of a mile. 



74 THE 20tli CENTURY ARITHMETIC. 

13. If a year is counted 365.25 days instead of the correct 
number, 365.242264 days, how great will be the error in 100 
years ? 

14. A mibic foot of ice weighs 57 pounds. If ice is selling 
at $8 a ton, what is a cubic foot worth ? 

15. A cubic foot of water weighs 1000 ounces. If this 
much water contains one pound of dirt, how much dirt is in 
a glassful (4.5 ounces) of water ? 

16. 32 dry quarts make a bushel, which contains 2150.42 
cubic inches. 4 liquid quarts make a gallon, which contains 
231 cubic inches. How many more cubic inches in the dry 
quart than in the liquid quart ? 

17. Find the cost of 78 bushels of wheat @ 62-^. 

18. At 87% ft how many knives can be bought for $14.40 ? 

19. What would 625 oranges cost at $2.37| per C.? 

20. What would 6225 bricks cost at $8.25 per M.? 

If the interest on $1 for 1 year is 6ft , for 1 month \ft , for 1 day 
i of a mill , what is the interest on 

21. $345.50 for 3 years 5 months 18 days ? 

22. $1443.25 for 4 years 7 months 12 days ? 

23. $785.85 for 3 years 6 months 6 days ? 

24. $1896.60 for 2 years 9 months 24 days ? 

25. $1776.45 for 1 year 11 months 18 days ? 

26. $1492.80 for 5 years 3 months 12 days ? 

200. Diversions.— i. Ten is added to a certain deci¬ 
mal. The separatrix is then moved one place to the left 
and ten is added. The sum is 4^- times the original deci¬ 
mal. What was the original decimal ? 

2. Find the difference between the smallest and largest 
decimals that can be expressed by the figures 8, 6, and 4. 

3. Ten times a certain decimal is .5. What is one tenth 
of the decimal ? 



DENOMINATE NUMBERS. 


75 


DENOMINATE NUMBERS. 

201. Quantity is measured by comparing it with a defi¬ 
nite amount of the same quantity. The definite quantity is 
chosen by law or custom, named, and called the unit; 
any larger quantity is a number of these units. 

The most commonly used units are the foot, the pound, the hour, 
and the gallon; with these the student should be perfectly familiar. 


DEFINITIONS. 

202. A denominate unit is that with which any quan¬ 
tity is compared. It is always concrete. 

One or more of these concrete units is called a denomi¬ 
nate number. I mile, $5, 9 pounds. 

A simple denominate number expresses units of but 

one denomination. 5 yards, 3 hours. (Denomination means a 
name.) 

A compound denominate number expresses units of two 
or more denominations of the same nature. 8 hours 15 min¬ 
utes ; 3 yards 2 feet 7 inches. 

Note. —This chapter teaches how to make calculations with denomi¬ 
nate numbers. The quantities themselves, the units of measure, must 
be learned by seeing and using them. A good way to teach the tables 
of measures would be to have the different measures labeled only 
as to name, and let the pupil learn their relative values by ex¬ 
perimenting and comparing. 

Estimating correctly the amount of any quantity depends entirely 
on familiarity with measuring it. Let the pupil make estimates of 
quantities, then measure and compare. This can easily be done with 
linear measure, applying it to the schoolroom and its furnishings. 

The difficulty in solving concrete problems is largely removed when 
the concrete terms are understood. The difficulty in making con¬ 
crete problems consists largely in the necessary estimate that must 
be made as to the pupil’s knowledge of things not at all arithmetical 
in character. 

Since arithmetic is the science of numbers , the author prefers the 
name denominate numbers to either compound quantities or measures. 



76 


DENOMINATE NUMBERS. 


203. The following diagram gives a general idea of the 
most important tables in denominate numbers. (For refer¬ 
ence.) 


I. Currency. 


II. Weights. 


III. Measures of 


l U. S. Money, 
j English Money. 

( Avoirdupois. Used for weighing most articles. 

< Troy. Used forweighing gold, silver and jewels. 

( Apothecaries’. Used in compounding drugs. 
r ( Lines. 

Extension. < Arcs. 

( Surfaces. 

C Volume. 

< Capacity. < Dry Measure. 

( Liquid Measure. 


Time. 


204. The following four tables are those in most com¬ 
mon use. The principles of all denominate numbers will be 
shown with them. 

Memorize the tables. 

I. Linear Measure. 

12 Inches = 1 Foot (ft.) 

3 Feet = 1 Yard (yd.) 

Yards = 1 Rod (rd.) 

320 Rods = 1 Mile (mi.). 

1 Mile=1760 Yards=5280 Feet=63360 Inches (in.). 

II. Avoirdupois Weight. 

16 Ounces == 1 Pound (lb.) 

100 Pounds = 1 Hundredweight (cwt.) 

20 Cwt. = 1 Ton (T.). 

1 Ton=20 Cwt.—2000 Pounds=32000 Ounces (oz.). 





DEFINITIONS. 


77 


III. Liquid Measure. 


IV. Dry Measure. 


4 Gills =1 Pint (pt.) 2 Pints =1 Quart (qt.) 

2 Pints =1 Quart (qt.) 8 Quarts=l Peck (pk.) 

4 Quarts=l Gallon (gal.). 4 Pecks =1 Bushel (bu.). 

1 Gallon=4 Quarts=8 Pints=32 Gills (gi.). 

1 Bushel=4 Pecks=32 Quarts=64 Pints (pt.). 


205. I. Linear Measure is used in measuring lengths 
and distances. 6 feet=l fathom. 8 furlongs=l mile. 

II. Avoirdupois Weight is used in weighing all articles 
except gold, silver, precious stones, and drugs when used in 
compounding medicines. 2000 pounds=the ton as com¬ 
monly employed. 2240 pounds=l long ton. The long ton 
is used at the U. S. custom-houses and in wholesale trans¬ 
actions of coal at* the mine and of iron ore. 

III. Liquid Measure is used in measuring liquids—milk, 
syrup, vinegar, etc. A gallon contains 231 cubic inches. A 
liquid quart therefore contains 231-^-4=57f cubic inches. 

IV. Dry Measure is used in measuring wheat, corn, po¬ 
tatoes, berries, and other dry articles. The bushel measure 
is 18£ inches in diameter and 8 inches deep; it contains 
2150.4 cubic inches. A dry quart therefore contains 2150.4 
=32=67^ cubic inches. 

6 times 67.2=403.2 7 times 57.75=404.25 

6 dry quarts=7 liquid quarts (nearly). 

The author deems it expedient to omit the numerous questions, 
such as How many inches in 1 foot? in 2ft.? how many pecks in 
1 bushel ? in 3 bu.,etc., which every teacher will have at her tongue’s 
end. Such questions, however, should be asked till the pupil is per¬ 
fectly familiar with the tables. 



78 


DENOMINATE NUMBERS. 


NOTATION. 

206. All lengths could be expressed in yards or fractions 
of a yard. But to avoid large numbers, we use rods and 
miles in expressing long distances; to avoid fractions, we 
use feet and inches in expressing short distances. We use 
hundredweights and tons to express heavy weights; pounds 
and ounces to express lighter ones. 

Abbreviations should not be pluralized. 5 pounds=5 lb.; 
9 inches=9 in.; 8 quarts=3 qt. 

a. 5 hundreds 2 tens 8 units=523 units. 

b. 5 dollars 2 dimes 3 cents=523 cents. 

C. 5 yards 2 feet 3 inches. 

If 10 inches made a foot and 10 feet a yard, c would equal 
523 inches; but the scale of compound quantities is not deci¬ 
mal and therefore this cannot be done. There is no shorter 
way of writing c than to abbreviate the words which name 
the units : 5 yd. 2 ft. 3 in. When there are no units of 

any denomination, that denomination is omitted. 4 yd. 7 in. 

The Metric System writes compound quantities decimally. 
(Art. 268.) 


REDUCTION. 

207. The reduction of a denominate number consists 
in changing its denomination without changing its value. 

Reduction Descending’ is changing an expression of 
quantity from a greater to a smaller unit of measure : 

Yards to inches, Gallons to pints, Tons to pounds. 

Reduction Ascending is changing an expression of quan¬ 
tity from a smaller to a greater unit of measure : 

Inches to yards, Pints to gallons, Pounds to tons. 



REDUCTION. 


79 


INTEGERS. 

Reduction Descending. 

208. Reduce 4 yd. to ft. 

Analysis.—I n 1 yd. there are 3 ft.; in 4 yd. 4 times 3 feet=12 ft. 


Reduce 

1. 4 yd. to ft. 

2. 3 yd. to ft. 

3. 2 ft. to in. 

4. 4 rd. to yd. 

5. 2 mi. to rd. 

6. 2 lb. to oz. 

7. 5 lb. to oz. 


22. Yd. toft.? 

23. Cwt. to lb.? 

24. Gal. to qt.? 


8. 2 cwt. to lb. 

9. 8 T. to cwt. 

10. 2 T. to lb. 

11. 8 pk. to qt. 

12. 5 bu. to pk. 

13. 3 bu. to pk. 

14. 2 pk. to qt. 

A ns. 

25. Ft. to in.? 

26. Bu. topk.? 

27. Pt. togi.? 


15. 7 qt. to pt. 

16. 2 gal. to qt. 

17. 5 gal. to qt. 

18. 2 pt. to gi. 

19. 3 qt. to pt. 

20 . 6 gal. to qt. 
21.5 mi. to yd. 

Multiply by 3. 

28. Lb. to oz.? 

29. Qt. to pt.? 

30. Rd. to yd.? 


How do you reduce yd. to ft ? 
How do you reduce 


209. 

Gal. 

5 

_4 

23 qt. 
_2 

47 pt. 


Reduce 5 gal. 3 qt. 1 pt. to pints. 

4 2 

qt. pt. Analysis.— In 1 gal. there are 4 qt.; in 

B 1 5 gal., 5 times 4 qt.=20 qt.; adding 3 qt. 

gives 23 qt. In 1 qt. there are 2 pt.; in 
23 qt., 23 times 2 pt.=46 pt.; adding 1 pt. 
gives 47 pt. Put the small figures above 
as shown. 

Linear Measure. 


Reduce 

1. 3 yd. 2 ft. to ft. 6. 2 yd. 2 ft. to in. 

2. 2 ft. 7 in. to iu. 7. 4 mi. 3 rd. 2 ft. to ft. 

3. 4 rd. 3 yd. to ft. 8 . 8 rd. 3 yd. 2 ft. to ft. 

4. 1 mi. 10 rd. to rd. 9. 3 rd. 5 yd. 1 ft. to ft. 

5. 1 mi. 1 ft. to ft. 10 . 8 mi. 30 rd. 4 yd. to yd. 



80 


DENOMINATE NUMBERS. 


Avoirdupois Weight. 


11 . 5 lb. 7 oz. to oz. 

12. 4 lb. 9 oz. to oz. 

13. 5 cwt. 8 lb. to lb. 


14 . 1 T. 11 cwt. to cwt. 

15 . 24 lb. 15 oz. to oz. 

Dry Measure. 


16. 8 T. 5 cwt. 75 lb. to lb. 

17 . 16 cwt. 90 lb. 12 oz. to oz. 

18. 5 T. 9 cwt. 80 lb. to lb. 

19 . 5 T. 19 cwt. to lb. 

20. 1 T. to lb. 



Reduce 


21 . 8 pk. 2 qt. to qt. 

22. 7 qt. 1 pt. to pt. 


26. 8 qt. 1 pt. to pt. 

27. 1 qt. 1 pt. 1 gi. to gi. 


23. 2 bu. 8 pk. 2 qt. to qt. 28. 2 gal. 8 qt. 2 pt. to pt. 


29. 5 gal. 2 qt. 1 pt. to pt. 

30. 1 gal. to gi. 


24. 8 bu. 5 qt. to pt. 

25. 1 bu. to pt. 


31 . 4 T. 2 cwt. 17 lb. to lb. Ans. 8217 lb. 

32. 6 rd. 4 yd. 2 ft. 6 in. to in. Ans. 1862 in. 

33. 6 bu. 2 pk. 6 qt. 1 pt. to pt. Ans. 429 pt. 

34. 5 cwt. 47 lb. 13 oz. to oz. Ans. 8765 oz. 

35 . 3 mi. 57 rd. 2 yd. 1 ft. to in. Ans. 201450 in. 

36. 23 gal. 3 qt. 1 pt. to pt. Ans. 191 pt. 

37 . 84 rd. 5 yd. 2 ft. 9 in. to in. Ans. 16845 in. 

38. 20 T. 16 cwt. 80 lb. 10 oz. to oz. Ans. 666890 oz. 

• 39. 16 gal. 3 qt. 1 pt. 3 gi. to gi. Ans. 543 gi. 

40. 25 bu. 2 pk. 7 qt. 1 pt. to pt. Ans. 1647 pt. 


Reduction Ascending. 


210. When there are more units of any denomination 
than will make one of the next higher, reduce to the next 
higher. 


4 8 2 

Bu. pk. qt. pt. 


Bu. pk. qt. pt. 


1 7 3 5 = 2 351 

Analysis. —Since 2 pt.=l qt., 5 pt.=2 qt. 1 pt. 

Since 4 pk. = l bu., 7 pk.=l bu. 3 pk. 



REDUCTION. 


81 


Reduce to higher denominations where possible : 


1. 1 bu. 7 pk. 8 qt. 5 pt. 

2. 2 bu. 5 pk. 2 qt. 7 pt. 

3. 5 bu. 11 pk. 1 qt. 9 pt. 

4. 4 bu. 1 pk. 27 qt. 5 pt. 

5. 8 bu. 19 pk. 18 qt. 1 pt. 

6. 8 gal. 9 qt. 7 gi. 

7. 1 gal. 11 qt. 7 pt. 

8. 15 qt. 4 pt. 7 gi. 

9. 19 gal. 15 pt. 

10. 2 gal. 8 qt. 11 pt. 


11. 1 yd. 18 ft. 9 in. 

12. 5J yd. 1 ft. 15 in. 

13. 11 yd. 1 ft. 18 in. 

14. 1 yd. 8 ft. 12 in. 

15. 640 rd. 2 yd. 7 ft, 

16. 1 T. 45 cwt. 18 lb. 

17. 87 cwt. 25 lb. 46 oz. 

18. 85 cwt. 80 lb. 90 oz. 

19. 8 T. 45 cwt. 60 oz. 

20. 6 T. 245 lb. 13 oz. 


211. Reduce 23 pints to higher denominations. 


Gal. 


4 

qt. 


Directions.—D ividing 23 by 2, the number 
of pints in a quart, we find that 23 pt.=ll qt. 
1 pt. Dividing 11 qt. by 4, we find that it 
equals 2 gal. 3 qt. 

# Arrange the work as shown, performing 

the divisions aside when necessary. Write the higher denomina¬ 
tions in the upper line only when it becomes necessary. 

Reduce to higher denominations : 


= 2 


11 

3 


pt. 

23 

1 

1 


Dry Measure. 

1. 23 pt. 

2. 75 pt. 

3. 19 qt. 

4. 64 pt. 

5. 36 pt. 

6. 105 pt. 

7. 236 pt, 

8. 134 pt. 

9. 973 qt. 
10. 1776 pt. 


Liquid Measure. 

11. 775 gi. 

12. 75 pt. 

13. 8250 gi. 

14. 450 pt, 

15. 784 pt. 

16. 54 pt. 

17. 725 gi. 

18. 1776 gi. 

19. 1492 gi. 

20. 287 pt. 


Avoirdupois Weight. 

21. 945 oz. 

22. 3420 1b. 

23. 8754 1b. 

24. 42684 oz. 

25. 64000 oz. 

26. 75000 1b. 

27. 80000 oz. 

28. 19000 1b. 

29. 2845 1b. 

30. 75000 oz. 



82 


DENOMINATE NUMBERS. 


212. In linear measure a difficulty sometimes arises in 
dividing by 5£. 5*=V, and to divide by V-, we multiply 

by 2 and divide by 11. 18^=18X*=-ff=8ft. In divid¬ 

ing 18 yd. by V 1 -, when we multiply by 2, it becomes not 86 
yd., but 86 /m^-yards. Consequently, when this 86 half¬ 
yards is divided by 11, the remainder is not 8 yd., but 8 
half-yards. 

Reduce 667 in. to higher denominations. 


534 3 12 

Rd. yd. ft. in. 

667 

=55 7 

=18 1 7 

=8 1*1 7 

1 6=* yd. 

8 117 


8 2 0 1 Ans. 


Directions. —Divide 667 by 12, 
then 55 by 3, then 18 by 534- The 
last remainder is 3 half-yards, 
which equals 134 yards. Reduce 
the 34 yard and add it as shown. 


Reduce to higher denominations : 

1. 667 in. 6. 8890 yd. 

2. 580 in. 7 . 63940 in. 

8. 78842 in. 


3. 77 ft. 

4. 1840 yd. 

5. 5287 ft. 


9. 17264 ft. 
10 . 980 in. 


11. 635 in. 

12. 700 ft. 

13. 1800 yd. 

14. 45000 ft. 

15. 10000 in. 


FRACTIONS. 

213. Reducing a Denominate Fraction to Integers. 


COMMON FRACTIONS. 


Reduce { 

5 X 

Rd. yd. 

§ = 4 
I X V ==f f = 4yV 
tj X3 = | = If 
1X12=9 


of a rod to integers of lower denominations. 


ft. 

1 


12 

in. 

9 Ans. 


rd. 


Analysis. —Multiplying £ rd. by 
it reduces it to yd. Multiplying 
the fractional part of yd. by 3 re¬ 
duces it to ft. Multiplying the 
fractional part of ft. by 12 reduces it 
to in. Begin the work by writing 
and put down the integers as obtained. 





REDUCTION. 


83 


Reduce to integers of lower denominations: 


1 . 

£ rd. 

9. 

2§ cwt. 

17. 

§A pk. 

25. 

2 rd. 3 

f yd. 

2 . 

f gal. 

10 . 

\ rd. 

18. 

qt. 

26. 

47f lb. 

3. 

£ bu. 

11 . 

tit gal. 

19. 

A ml- 

27. 

9 bu. 

2* pk. 

4. 

A lb. 

12 . 

7f bu. 

20 . 

AT. 

28. 

5 mi. 

2f rd. 

5. 

f yd. 

13. 

T2 T. 

21 . 

t 6 t rd. 

29. 

7 mi. 

5§ yd. 

6 . 

5§ bu. 

14. 

t mi. 

22 . 

A bu. 

30. 

8 T. 37#f cwt. 

7. 

A t. 

15. 

4y 5 r gal. 

23. 

4 t 3 t mi. 

31. 

7/rT. 


8 . 

t mi. 

16. 

2f rd. 

24. 

A mi. 

32. 

9§ mi. 



DECIMAL FRACTIONS. 


Reduce .8282 T. to integers of lower denominations. 


T. 

.8232 = 
20 

16.4640 

46.40 

16 

6.4 


wt. lb. oz. Analysis. -Multiplying .8232 

16 46 6.4 Ans. ^ ^ reduces it to cwt. 

Multiplying the decimal part 
of the product by 100 reduces 
it to lb. Multiplying the dec¬ 
imal part of this product by 16 reduces it to oz. 

Write first .8232 T. = , then the integers as they 
are obtained. 


Reduce to integers of lower denominations : 


1. .8 bu. 

2. .9 gal. 

3. .75 yd. 

4. .875 cwt 

5. .045 bu. 

6. .683 T. 

7. 8.24 gal. 

8. 2.65 pk. 


9. .0755 T. 

10. .0875 mi 

11. 2.435 rd. 

12. .078 cwt. 

13. .005 bu. 

14. .008 mi. 

15. .0007 T. 

16. .37 bu. 


17. .013 bu. 

18. .83 rd. 

19. .0031 T. 

20. .084 mi. 

21. .035 gal. 

22. .0183 T. 

23. .6813 rd. 

24. .871 bu. 


25. .1776 T. 

26. .1776 mi. 

27. .1776 cwt, 

28. .1776 gal. 

29. .1492 rd. 

30. .1492 mi. 

31. .1492 T. 

32. .1492 bu. 






84 


DENOMINATE NUMBERS. 


To Reduce Lower Denominations to a Fraction of a 
Higher Denomination. 


215. Reduce 8 yd. 2 ft. 7 in. to a fraction of a rod. 


5^ 

Rd. yd. 

8 

= 8 
= 8J* 

==»! 

=.7+ Ans. 


8 12 

ft. in. 

2 7 

2 t 7 ? 


Analysis. —Dividing in. by 12 re¬ 
duces them to ft. Dividing ft. by 3 
reduces them to yd. Dividing yd. 
by 5 % reduces them to rd. 

The common fraction may be re¬ 
duced to a decimal. 


Reduce to a fraction of the higher denomination : 


1. 8 pk. to bu. 

2. 1 pt. to gal. 

3. 25 lb. to T. 

4. 9 in. to yd. 

13. 18 cwt. 85 lb. 8 oz. to T. 

14. 95 rd. 4 yd. 2 ft. to mi. 

15. 8 yd. 2 ft. 9 in. to rd. 

16. 60 lb. 8* oz. to cwt. 


ft. to rd. 

10. | yd. to rd. 

11 . 8 qt. 1 pt. to gal- 

12. 2 pk. 5qt. to gal. 

17. 8 pk. 6 qt. 1 pt. to bu. 

18. 9 rd. 5yd. 2 ft. 11 in. to mi. 

19 . 18 cwt. 90 lb. 14 oz. to T. 

20. 185 rd. 1 ft. 10 in. to mi. 


5. 100 rd. to mi. 9. £■ 

6. * pt. to gal. 

7. | qt. to bu. 

8- TO oz. to cwt. 


216. ADDITION. 

Only like quantities can be added. 

What is the sum of 1 gal. 8 qt. 1 pt. 8 gi., 2 qt. 2 gi., 
5 gal. 1 qt. 1 pt. 3 gi., and 2 gal. 3 qt. 1 pt. 2 gi.? 

Directions. — Place numbers of the 
same denominations under each other. 
Add each column, then reduce to higher 
denominations where possible. 


Gal. qt. 

1 8 

2 

5 1 

2 8 


2 4 

pt. gi. 

1 8 

2 

1 8 

1 2 


8 9 8 10 

10 3 1 2 Ans. 




ADDITION. 


85 


Find the sum of the following: 


1. Gal 


qt. 

pt. 

2. Bu. 

pk. 

qt. 

pt. 

3. T. 

cwt. 

lb. 

oz. 

1 


2 

1 

6 

3 

7 

1 

5 

16 

73 

11 

2 


3 

1 

5 

2 

4 

1 

2 

12 

18. 

15 

4 


1 

1 

rj 

i 

3 

5 

1 

1 

9 

47 

8 

4. Rd. 

yd. 

ft. 

in. 

5. Mi. 

rd. 

yd. 

ft. 

6. Bu. 

pk. 

qt. 

pt. 

6 

2 

2 

7 

7 

145 

4 

1 

18 

3 

7 

1 

5 

3 

1 

9 

8 

284 

3 

2 

15 

2 

5 

1 

7 

4 

2 

11 

15 

300 

5 

2 ' 

17 

1 

3 

1 

7. T. cwt. 

lb. 

oz. 

8. Mi. 

rd. 

yd. 

ft. 

9. Bu. 

pk. 

qt. 

pt. 

3 

18 

83 

12 

6 

180 


2 

4 

3 

1 

17 

15 


9 

3 

300 

5 

1 

35 

3 

7 


4 

9 

76 



245 


2 


2 

5 

1 

5 

13 


15 

1 


3 


19 

1 


1 

io. 7 

gal. 3 qt. 1 

pt., 9 gal. \ 

2 qt. 

8 gi 

., 5 gal 

. 2 qt. 1 

pt. 

2 gi., and 8 

qt. 

1 pt. 

8 gi. 








11. 5 T. 18 cwt. 85 lb. 7 oz., 7 T. 9 cwt. 15 oz., 9 T. 67 lb. 
18 oz., and 15 T. 78 lb. 11 oz. 


12. 75 mi. 180 yd. 28 in., 84 mi. 1850 yd. 35 in., 11 mi. 
1545 yd. 81 in. 

13. 25 bn. 8 pk. 6 qt., 17 bn. 2 pk. 1 pt., 84 bu. 7 qt. 1 pt., 
and 4 bu. 8 pk. 5 qt. 1 pt. 

14. 81 mi. 128 rd. 4 yd., 75 mi. 285 rd. 5 yd., 815 rd. 
4 yd. 2 feet., and 65 mi. 228 rd. 8 yd. 2 ft. 

15. 16 mi. 220 rd. 5 yd., 9 mi. 200 rd. 8 yd., 65 rd. 3 yd. 

1 ft. 9 in., and 13 mi. 5 yd. 2 ft. 10 in. 

* 


217. SUBTRACTION. 

Only like quantities can be subtracted. 

From 7 bu. 2 pk. 5 qt. subtract 5 bu. 3 pk. 2 qt. 












86 


DENOMINATE NUMBERS. 


Bu. 

7 

5 


4 

pk. 

2 

8 


8 

qt. 

6 

2 


Directions. —Place like denominations un¬ 
der each other and, beginning on the right, 
subtract. 3 pk. can not be subtracted from 

_ 2 pk. ; therefore, take 1 bu. (= 4 pk.) from 

1.8 4 Ans. 7 bu., add the 4 pk. and 2 pk., then subtract 

3 pk. from 6 pk. Having taken 1 bu. from 7 bu., subtract 5 from 6. 
1 . 


4. 


Bu. 

P k. 

qt. 

pt. 

2. T. 

cwt. 

lb. 

oz. 

3. 

Mi. yd. 

ft. 

in. 

18 

8 

7 

1 

16 

18 

84 

15 


7 

845 

2 

11 

18 

2 

5 

1 

9 

8 

62 

7 


2 

430 

2 

7 

Bu. 

pk. 

qt. 

pt. 

5. Rd. 

yd. 

ft. 

in. 

6. 

T. 

cwt. 

lb. 

oz. 

7 

2 

5 


200 

4 

1 

9 


5 

8 

54 

9 

2 

8 

1 

1 

100 

2 

2 

11 


2 

10 

41 

13 

Gal. 

qt. 

pt. 

gi* 

8. Mi. 

rd. 

yd. 

ft. 

9. 

T. 

cwt. 

lb. 

oz. 

17 

1 

1 

2 

25 

14 

4 

1 


91 

4 

85 

3 

9 

8 


8 

18 

180 

1 

2 


18 

15 

47 

9 


10. From 11 bu. 1 pk. 7 qt. subtract 5 bu. 2 pk. 4 qt. 

11 . From 55 mi. 4 yd. 2 ft. subtract 16 mi. 8 rd. 5 yd. 

12. From 17 T. 1500 lb. 9 oz. subtract 9 T. 18 cwt. 11 oz 

13. From 18 gal. 1 qt. 1 pt. subtract 7 gal. 8 qt. 8 gi. 

14. From 18 mi. 120 rd. 1 ft. subtract 12 mi. 5 yd. 2 ft. 

15. From 27 mi. 8 yd. 9 in. subtract 17 mi. 200 rd. 2 ft. 


218. MULTIPLICATION. 

The multiplier of a concrete number must be abstract. 
Multiply 2 bu. 8 pk. 5 qt. 1 pt. by 7. 

Directions. —Place the multiplier un¬ 
der the lowest denomination and begin 
on the right to multiply. Place the 
products under the proper denomina¬ 
tions, then reduce them to higher 
1 Ans . denominations. 


4 8 2 

Bu. pk. qt. pt 

2 8 5 1 

7 

~14 21 85 7 














DIVISION. 87 


1 . 

Bu. 

6 

pk. qt. 

1 7 

pt. 

1 

8 

2. 

Gal. qt. 

5 2 

pt. gi. 

1 8 

7 

3. 

T. lb. 

4 1850 

oz. 

7 

5 

4. 

Mi. 

yd. 

ft. 

5. 

Rd. 

yd. 

in. 

6. 

Pk. qt. 

pt. 


4 

420 

2 


100 

5 

6 


2 5 

1 




6 




9 



8 

7. 

Rd. 

yd. ft. 

in. 

8. 

Cvvt. 

lb. 

oz. 

9. 

Rd. yd. 

ft. 


800 

4 2 

7 


9 

80 

10 


70 5 

2 




2 




9 



40 


Multiply 

10. 11 bu. 1 pk. 7 qt. by IB. 13. 18 gal. 1 qt. 1 pt. by 82. 

11. 55 mi. 4 yd. 2 ft. by 87. 14. 18 mi. 120 rd. 1 ft. by 64 

12. 17 T. 1500 lb. 9 oz. by 45. 15 . 27 mi. 8 yd. 9 in. by 98. 


219. DIVISION. 

Divide 22 gal. 8 qt. 1 pt. 8 gi. by 5. 




4 

2 

4 





Gal. 

qt. 

pt. 

gi. Directions.- 

—Place the 

divisor as 

5 

| 22 

8 

1 

3 shown a 

nd begin at the left to divide. 



8 

2 

19 5 is contained 

in 22 gal. 4 times, with a 





— remainder of 2 gal. This 2 gal. has not 



11 

8 

yet been divided. Reduce it to qt. 


4 

2 

0 

3 Ans. (multiply by little figure over qt.), add 





the 8 to 3, divide 11 by 5, and so on. 

l. 

Bu. 

pk. 

qt. 

2 . Gal. qt. 

pt. 

3. Bu. 

pk. qt. 


2 [_6 

8 

2 

3 |_9 1 

1 

5 |_26 

3 7 

4. 

T. 

cwt. 

lb. 

5. Rd. yd. 

ft. 

6. Mi. 

rd. yd. 


6 |_11 

18 

84 

4 | 6 2 

2 

8 |_84 

62 5 

7. 

Mi. 

rd. 

yd. 

8. T. lb. 

oz. 

9. Yd. 

ft. in. 

60 

» | 280 

800 

4 

154 | 8 1600 

9 

16 | 780 

2 11 
















88 


DENOMINATE NUMBERS. 


Divide 

10. 86 bn. 8 pk. 7 qt. 1 pt. by 9. 

11. 18 mi. 145 rd. 5 yd. 2 ft. by 8. 

12. 87 T. 18 cwt. 95 lb. 14 oz. by 25. 

13. 15 gal. 3 qt. 1 pt. 2 gi. by 11. 

14. 64 mi. 90 rd. 4 yd. 2 ft. by 35. 

15. 315 rd. 5 yd. 2 ft. 11 in. by 18. 


220. REVIEW. 

Reduce 

1. 3 mi. 150 rd. 4 yd. 2 ft. to ft. 

2. 3 T. 17 cwt. 85 lb. 12 oz. to oz. 

3. 7 gal. 3 qt. 1 pt. 3 gi. to gi. 

4. 5 bu. 2 pk. 7 qt. 1 pt. to pt. 

Reduce to higher denominations : 

5. 1000 pt. 6. 500 gi. 7. 10000 oz. 8. 50000 in. 

Reduce to integers of lower denominations : 

9. H b u. 10. 4 7 5 T. n. 3.1416 bu. 12 . 1.6093 T. 

13. T is mi. 14. t 9 t gal. 15. .3937 mi. 16. 1.4142 gal. 

Reduce to a fraction of the higher denomination : 

17. 25 yd. to fraction of a mi. 19. 77 lb. to fraction of a T. 

18. 3 pt. to fraction of a gal. 20. 7 qt. to fraction of a bu. 

21. Add 1 T. 3 cwt. 55 lb. 11 oz., 17 cwt. 36 lb. 12 oz., 
4 T. 13 cwt. 15 oz., and 3 T. 85 lb. 12 oz. 

22. From 3 mi. 185 rd. 3 yd. take 1 mi. 250 rd. 9 in. 

23. Divide 25 bu. 3 pk. 6 qt. 1 pt. by 6. 

24. Multiply 2 gal. 3 qt. 1 pt. 2 gi. by 9. 

25. A car-load of oranges was sold for $600 at the rate of 

10/ a dozen. If there were 12 doz. in a box, how many 

boxes were there ? 



TROY WEIGHT. 


89 


221. V. Troy Weight. 

24 Grains =1 Pennyweight (pwt.) 

20 Pennyweights=l Ounce (oz.) 

12 Ounces =1 Pound (lb.). 

1 Pound=12 Ounces=240 Pennyweights=5760 Grains (gr.). 

Troy Weight is used in weighing gold, silver, precious 
stones, and in a few scientific operations. 

222. United States coins expressed in this weight are as 
follows : The gold dollar, 25.8 grains. The silver dollar 
412| grains. The small silver coins, 885.8grains to the dollar 
—that is, 10 dimes, 4 quarters, or 2 half-dollars weigh 885.8 
grains. The gold and silver coins are .9 pure metal and .1 
alloy. The metal as coined—that is, with the alloy in it—is 
called standard gold or silver. The nickel 5/ piece, 77.16 
grains. The nickel 3^ piece, 30 grains. The copper 1/ 
piece, 48 grains. 

In weighing diamonds, 1 carat = 3£ grains. 

Carat, applied to gold or.silver jewelry, indicates the num¬ 
ber of parts in 24 that are pure gold or silver. 18 carat = 
pure gold or silver. 

Problems. 

1. How many oz. in 4 lb. 3 oz.? 

2. How many pwt. in 5 lb. 7 oz. 13 pwt.? 

3. If the output of a Colorado silver-mine averages 590 
oz. daily, what is the value of the production in thirty days 
at $8 per lb.? 

4. How many silver dollars can be made from 10 lb. of 
pure silver ? 

5. How many gold dollars can be made from 10 lb. of 
pure gold ? 

6. Find the total weight of pure gold in 7 double eagles 
($20 gold pieces), 7 eagles ($10 gold pieces), 9 five-dollar 
gold pieces, and 6 one-dollar gold pieces. 


90 


DENOMINATE NUMBERS. 


7. What weight of silver can I purchase for $1000 when 
1 oz. of silver is worth 59£/? 

8. How many five-dollar gold pieces can be made from 
100 lbs. of standard gold ? 

9. How much pure gold is in 10 lb. of 18 carat gold 
rings ? 

10. How many silver dollars could be made from 240 lb. 
of 14 carat family silver plate ? 


223. Comparison of Troy and Avoirdupois Weights. 


A Troy pound is lighter than an Avoirdupois pound, but a 
Troy ounce is heavier than an Avoirdupois ounce. The grain 
is the only denomination that is the same in the two weights; 
hence, all comparisons should be made by reducing the 
given quantities to grains : 


Troy. 

5760 grains= 1 lb. Troy. 

1 grain = 5 -i 5!r lb. Troy. 
7000 graius= lb. Troy. 
1 Avoir. lb.= |ff lb. Troy. 


Avoirdupois. 

7000 grains= 1 lb. Avoir. 

1 grain lb. Avoir. 

5760 grains=^§° lb. Avoir. 
1 Troy lb. = lb. Avoir. 


Problems. 

1. How many Troy lb. in 10 Avoir, lb.? 

2. How many Avoir, lb. in 10 Troy lb.? 

3. If a man buys gold at $100 per Avoir, lb. and sells it at 
$100 per Troy lb., what is his gain on 72 Avoir, lb.? 

4. If a man buys gold at $100 per Troy lb. and sells it 
at $100 per Avoir, lb., what is his loss on 72 Troy lb.? 

5. A miner sells $8000 worth of gold at $16 an ounce 
Troy. What was the weight Avoirdupois ? 

6. Reduce 21 lb. 14 oz. Avoir, to lb., oz., etc., Troy. 

7. What part of an ounce Troy is an ounce Avoirdupois ? 



APOTHECARIES' WEIGHT . 


91 


8 . How many gold dollars can be made from 10 lb. Avoir, 
of 12 carat gold ? 

9. Which is heavier, 1 lb. of gold or 1 lb. of sugar ? 1 oz. 

of gold or 1 oz. of sugar ? 

10 . Reduce 7500 ounces of gold and 7500 ounces of sugar 
to pounds. 

11 . What is the weight in lb. Avoir, of 5000 8 -carat 
diamonds ? 


VI. Apothecaries’ Weight. 

20 Grains =1 Scruple (s) 

8 Scruples=l Dram ( 3 ) 

8 Drams =1 Ounce ( 5 ) 

12 Ounces =1 Pound (lb), 
lb 1=5 12=3 96=D 288=gr. 5760. 

Apothecaries’ Weight is used by apothecaries in com¬ 
pounding drugs. Physicians, therefore, write their prescrip¬ 
tions in this weight. 

Notice that the abbreviations precede the numbers. 

The pound, ounce, and grain of this weight are identical 
with those of Troy weight; therefore, the comparison of 
Apothecaries’ and Avoirdupois weights is the same as that of 
Troy and Avoirdupois weights. 

Druggists buy medicines in bulk by Avoirdupois and sell 
them by Troy weight. 

Physicians write their prescriptions in Roman notation, 
using j for final i. 

Read : 

1 . Ibv. 5 vij .3 v. 3 ij. gr. xij. 2 . 5 iv .'3 iv. aij. 

3- I iij- 3vij. 3ij.gr. xviij. 4. 3 vj. 9 ij. gr. xvj. 

5. A druggist buys an Avoir, lb. of quinine for $5 and 
sells it at 50/ per oz. Apoth. What is his gain ? 



92 


DENOMINATE NUMBERS . 


6. A druggist buys 1501b. camphor for $16 and sells it 
at 40/ per 3 . What, is his total gain ? 

7. Reduce 5 lb. Avoir, to Apoth. weight. 

8. A druggist buys 20 Avoir, lb. of a drug at $8 per lb. and 
sells it at 25/ per dram. What is his total gain ? 

9 . Mr. Druggist buys 250 Avoir, lb. of borax for $50. At 
what must he sell it per lb. Apoth. to gain $50 ? 

10. How many 5-grain capsules can be filled from 3 xj. 
3. vi Q ij. gr. x. quinine ? 


225. VII. Numbers. 

12 Units = 1 Dozen. 

12 Dozen = 1 Gross. 

12 Gross — 1 Great Gross. 
20 Units == 1 Score. 


VIII. Paper. 

24 Sheets = 1 Quire. 
20 Quires = 1 Ream. 

2 Reams = 1 Bundle, 
5 Bundles = 1 Bale. 


1 Great Gross=12 Gross=144 Dozen—1728 Units. 


This table is used in counting eggs, spools of thread, but¬ 
tons, and other articles that are reckoned by the piece, and 
not by weight or length. 

1 Bale=5 Bundles—10 Reams=200 Quires=4800 sheets. 

This table is used in counting paper. As sheets are 
usually doubled, it takes 48 single sheets to make a quire. 


Problems. 


1. How old is a Bible scholar who is threescore years and 
ten ? 

2. If 10 eggs weigh a pound (cook-book recipes), how many 
eggs in 170f pounds ? 

3. How many buttons on 100 cards, each card having 4 
rows, 9 buttons to the row ? 

4. How many sheets in a ream ? 



ENGLISH OR STERLING MONEY. 


98 


5. At $2.50 a ream, how many whole sheets could be ob¬ 
tained for 25/ ? 

6 . At $25.92 a great gross, what must thread be sold at 
per spool to make a total profit of $25.92 ? 


226. IX. English, or Sterling Money. 

4 Farthings = 1 Penny (d.) 

12 Pence = 1 Shilling (s.) 

20 Shillings =1 Pound (£). 

1 Pound=20 Shillings=240 Pence=960 Farthings (far.). 

English money is used in England, Scotland, Ireland, and 
most of the English possessions. Before 1776, it was the 
legal money of this country, and was in actual circulation 
to some extent as late as 1860. 

A pound, called also a sovereign, is equal to $4.8665. The 
English also have a crown = 5 shillings, and a florin — 2 
shillings. 

Farthings are usually written as fractions of a penny. 

3 pence 1 farthing = 3l^d. 9 pence 3 farthings = 9%d. 

The gold coins, sovereign and half-sovereign, are pure 
gold and alloy. The silver coins are f$ pure silver and 
? 3 o alloy. The metal as coined—that is, with the alloy in 
it—is called standard gold or silver. 

Problems. 

1 . Add £7 18s. 5d., £24 15s. 9d., and £17 IBs. 4d. 

2 . Multiply £4 7s. 9d. by 8 . 

3 . From £75 8 s. Id. subtract £50 17s. 19d. 

4 . Divide £14 10s. 5d. by 8 . 

5 . Reduce 785d. to higher denominations. 

6 . Reduce 9s. 7d. to the decimal of a £. 

7 . Reduce £ x 9 g- to integers of lower denominations. 



94 


DENOMINATE NUMBERS. 


8. 480 oz. Troy of standard gold is coined into 1869 
sovereigns. How many grains of standard gold in a £ ? 

9. 12 oz. Troy of standard silver is coined into 66 shil¬ 
lings. How many grains of standard silver in a shilling ? 

10. 1 lb. Troy of standard silver is coined into 66 shil¬ 
lings. If 512 crowns weigh 82 lb. Avoir., what is the error in 
the weight of 1 crown ? 

11. If a man buys 4 horses for £100, and sells them for 
£98 10s. 6d., what is the loss on each horse? 

12. The price of gold is £8 17s. 10|d. per oz. What is 
the least number of ounces that will make an exact num¬ 
ber of sovereigns ? (No alloy considered.) 

What is the value in 

13. U. S. money of £9.788? 

14. U. S. money of £17 12s.? 

15. U. S. money of £250 18s.? 

16. U. S. money of 15s. 9d.? 

17. English money of $8898.20 ? 

18. English money of $1000 ? 

19. English money of $8425 ? 

20. English money of $500 ? 

227. Square or Surface Measure. 

A flat surface having straight sides and four square corners 
is called a rectangle. Each page of this book is a rectangle. 

a- When the sides are equal in length it is called a 

square. 

Square measure is used in measuring areas. 

Areas are not measured directly but are computed. 

This rectangle is 4 inches long and 8 inches broad, and its 
surface is divided into square inches. Its area is seen to 
consist of 8 rows with 4 squares in each row; therefore, it is 
8 times 4 square inches. 



SQUARE OR SURFACE MEASURE. 


95 



The area of a rectangle is equal to the product of its length and 
breadth. 

b. The length and breadth must be expressed in the same 
unit, both inches, both feet, etc.; otherwise, the small fig¬ 
ures formed by the cross lines will not be squares. The area 
will be squares of the same unit. It is not correct, how¬ 
ever to say “3 in. times 4 in.=12 square inches.” The 
analysis shows that the area is 3 times 4 square in.=12 
square in. Concrete numbers can not be multiplied together. 

The length of a rectangle is always its longer, and the 
breadth its shorter, dimension. The dimensions of the rect¬ 
angle above may be briefly expressed 3 // X4 // , in which X 
is read “by” and " inches. ' is the sign for feet. The 
product of the numbers is the number of small squares. 









96 


DENOMINATE NUMBERS. 


What is the area of a rectangle whose dimensions are 


1 . 

2 . 


3. 


4. 


8"X4"? 5. 2'X9' ? 9. 6"Xll"? 

2"x7"? 6. 3'X7'? 10 . ,6"X 13" ? 


5"X9"? 7. 4'x8'? 11 . 9"X15"? 
6"Xll" ? 8. 6'X9'? 12. 7"Xl7"? 

17. 12"X12"? What is 12 inches called? 


13. 12'X19'? 

14. 15'X21'? 

15. 16'X48'? 

16. 24'X58'? 
What is the 


figure whose dimensions are 12"Xl2"? Ans. 1 square foot. 
18. 3'X3'? What is 8 feet called ? What is the figure in 


this question ? 

19. 5£ yd.x5£ yd.? What is 5£ yd. called? What is 
the figure in this question ? 

20. The square on % an inch is what part of the square on 
an inch ? Draw the squares. 


228. X. Table. 

12 2 or 144 Square Inches= 1 Square Foot (sq. ft.) 

8 2 or 9 Square Feet = 1 Square Yard (sq. yd.) 
(5£) 2 or 80^ Square Yards = 1 Square Rod (sq. rd.) 
160 Square Rods = 1 Acre (A.) 

640 Acres = 1 Square Mile (sq. mi.). 

Square measure is used in measuring areas. 

1 acre=a square measuring 208.71+ft. on each side. 

1 acre=48560 square ft. 

A square rd. is also called a perch. 40 sq. rd.=l rood. 
A square of flooring or roofing=100 sq. ft. 

A section of land, U. S. survey=l square mile. 

A township=86 square miles. 

Reduce 

1. 5 A. to sq. ft. How many sq. ft in 1 A. ? 

2. 27480 sq. ft. to higher denominations. 

3. 78284 sq. yd. to higher denominations. 

4. f of a sq. mi. to lower denominations. 

5. | of a sq. mi. to lower denominations. 

6. 68 A. 15 sq. rd. 22 sq. yd. to sq. ft. 


SQUARE OR SURFACE MEASURE. 


97 


7. What decimal part of a sq. mi. is 800 A. 100 sq. rd. ? 

8. What part of an acre is 10890 sq. ft. ? 

9. What part of a township it 7000 A. ? 

10. What is the difference between 16 square feet and a 
square 16'X16' ? 

11. If an acre produces 80 bushels of wheat, how much 
will a rectangular farm 500 yd. X900 yd. produce ? 

12. Tennis courts are 36'x78'. What decimal part of an 
acre is its area ? 

13. How many brick, laid lengthwise, whose surfaces are 
6"X2" are in one block of a street 600'X 50' ? 

14. a. A grass-cutter is 2 ft. wide. How often must it be 
pushed lengthwise a lawn which is 80'X 60' to cut all the 
grass on it ? b. What would be the area of the part cut if 
it were pushed once around the outer edge ? (Diagram.) 

15. What is the area (part of an A.) of a baseball dia¬ 
mond ? The diamond is a square, 90' on each side. 

16. A cricket-field 100 yd. long, 57 yd. wide, has a path 5 
feet wide going all around it just inside the outer lines. 
W hat is the area of the path ? (Draw a cricket-field.) 

17. A football field is 330'X120'. What decimal part of 
an acre is its area ? 

18. How many acres were covered by the Manufactures 
and Liberal Arts Building at the World’s Fair? It 
measured 1687 ft. long, 787 feet wide. 

19. The base of the largest Egyptian pyramid, Cheops, is 
a square 693 ft. each way. How many acres has it for a 
base ? 

20. Solomon’s temple was 60 cubits long and 20 cubits 
wide. If a cubit was 20 inches, how many square feet was 
its area ? 


98 


DENOMINATE NUMBERS. 


229. If the area of a rectangle is 80 sq. ft. and its length 
is 5 ft., what is its breadth ? 5 times what=80 ? 

If the area and one dimension of a rectangle are given, 
how do yon find the other dimension ? 

Ans. By dividing the area by the given dimensions. The 
square and linear units must be the same denomination. 


What is the other dimension 
(draw the figure) 

1. 20 sq. ft. and length 5 ft.? 6. 

2. 21 sq. ft. and length 7 ft.? 7. 

3. 18 sq. ft. and length 6 ft.? 8. 

4. 85 sq. ft. and length 7 ft.? 9. 

5. 82 sq. ft. and length 8 ft.? 10 . 

11. An acre contains 48560 sq 
taining f of an acre is 121 ft. 
other side ? 

12. A rectangle containing £ 
side. What is the other side ? 


of a 7'ectangle ivhose area is 

80 sq. ft. and breadth 5 ft.? 
28 sq. ft. and breadth 4 ft.? 
42 sq. ft. and breadth 6 ft.? 
45 sq. ft. and breadth 5 ft.? 
54 sq. ft. and breadth 6 ft.? 
.ft. A rectangular lot con- 
on one side. What is the 

of an acre is 800 ft. on one 


230. Cubic Measure. 

There are six faces to this book : the two backs, the two 
ends, the hinge face and the opening face. How many faces 
has a brick ? A figure like a brick is called a rectangular 
parallelopiped : it has six rectangular faces. 

When all the faces are equal squares the figure is called a 

cube. 

Cubic measure is used in measuring volumes. Volumes 
are not measured directly, but are computed. This paral¬ 
lelopiped is 5 inches long, 4 inches broad, and 8 inches 
thick. Its volume is seen to consist of 8 layers with 20 
cubic inches in each layer ; therefore, its volume is 8 times 
20 cubic inches. 





CUBIC MEASURE. 


99 



The volume of a parallelopiped is equal to the product of its 
length , breadth , and thickness. 

The dimensions must be expressed in the same unit, all 
inches, all feet, etc., otherwise, the small figures formed by 
the cross-cuts would not be cubes. The volume will be 
cubes of the same unit. It is not correct, however, to say 
8 in.X4 in.X5 in.=60 cubic inches. The analysis shows 
that the volume is 8 times 20 cubic inches=60 cubic inches. 
Concrete numbers cannot be multiplied together. The 
dimensions of the parallelopiped above may be briefly ex¬ 
pressed 8 // X4 // X5". The product of the numbers is the 
number of small cubes. 

What is the volume of a parallelopiped whose dimensions are 

1. 8"X4"X5"? 4. 8'X7'X5'? 7. 9"Xl5"X 12" ? 

2 . 2"X7"X8"? 5. 4'X8'X8'? 8 . 7"X 17"X 10" ? 

3. 5"X9"X4"? 6. 6'X9'X4'? 9. 12"X 19"X 11" ? 



























100 


DENOMINATE NUMBERS. 


10 . 6"Xll"X8"? 12 . S'Xll'XO'? 14. 15"X21"X18"? 

11. 2"X9"X5"? 13. 6'Xl8'X5'? 15. 16"X43"X20" ? 

16. 12"Xl2"Xl2"? What is 12 inches called ? What is 
the figure in this question ? 

17. 3'X3'X3'? What is 3 feet called? What is the 
figure in this question ? 

231. XI. Table. 

12 3 or 1728 Cubic Inches = 1 Cubic Foot (cu. ft.) 

3 3 or 27 Cubic Feet = 1 Cubic Yard (cu. yd.). 

1 Cubic Yard=27 Cubic Feet=46656 Cubic Inches (cu. in.). 

Cubic Measure is used in measuring volumes. 

The bushel measure is round, 18| inches in diameter and 
8 inches deep. It contains 2150.4 cubic inches. 

The gallon contains 231 cubic inches. 

A dry quart contains 2150.4^-32=67^ cubic inches. 

A liquid quart contains 231-^-4=57 j cubic inches. 

Problems. 

1 . How many gallons in a cubic foot ? 

2. How many bushels in a cubic yard ? 

3 . How many gallons will a vessel hold whose dimensions 
are 7"X8"X20"? 

4 . How many cubic yards of dirt were removed from a 
cellar, the floor of which is 12'Xl6' and height 7' ? 

5. What part of a cubic foot is in a brick 8"X4"X2" ? 

6. How many gallons are in a bathtub 5'X2'X2 / , if the 
tub is half full ? 

7. How many cu. ft. of wheat in a bin 4'X4'X3' ? 

8. How many cubic feet of air to each of 30 pupils in a 
room 20 ft. square and 12 ft. high ? 


VOLUME AND WEIGHT. 


101 


232. Volume and Weight. 


60 lb. Wheat=l Bushel. 
56 lb. Corn =1 Bushel. 
82 lb. Oats =1 Bushel. 
80 lb. Coal =1 Bushel. 


56 lb. Butter=l Firkin. 
100 lb. Nails =1 Keg. 
196 lb. Flour =1 Barrel. 
200 lb. Beef =1 Barrel. 


Problems. 


How many bushels of wheat can be put in a bin 


l. 4 / x4 / x3 / ? 


2 . 

3'X5'X2'? 


How many bushels 

of 

corn can 

be put in a 

bin 

3. 4 / X4 / X3 / ? 


4. 



How many bushels 

of 

oats can 

be put in a 

bin 

5'. 4'X4 / X3 / ? 


6 . 

8'X5'X2'? 



7. At the rate of 25/ a bushel, what is coal worth per ton ? 

8. How many firkins of butter in 1400 lb. ? 

9. How many 6-penny nails, at 80 nails to the lb., are in 
a keg ? 

10. If a barrel of flour is worth $5, what will a 10/ loaf 
of bread weigh when the baker receives for bread $5 more 
per barrel of flour than it costs ? 

11. How many barrels of beef will a freight-car carry, if 
its capacity is 40 tons ? 

12. A bushel of corn meal weighs 48 lb. If a miller 
gives meal for corn, bushel for bushel, how much does he 
make on 100 bushels of corn @ 40/ per bushel ? 

13. If a cubic foot of water weighs 1000 oz. Avoir., how 
many pounds does it weigh ? 

14. What is the weight of 1 cu. in. ? (Decimal to 3d order.) 

15. What does a liquid pint (28J cu. in.) weigh ? 

16. How many cubic feet in 1 ton of water ? 

17. If a cubic foot of ice weighs 57.5 lb., how many tons 
can be put in an ice-house 85'X 15'X 10' ? 

18. A cubic foot of gold weighs 1204 lb. What is the 
weight of 100 cubic inches ? 


102 


DENOMINATE NUMBERS. 


19. A cubic foot of silver weighs 626 lb. What is the 
weight of 100 cubic inches ? 

20. If a cubic foot of water weighs 62.5 lb., what is the 
weight of 1 gallon ? 


233. Wood Measure. 

Any pile of wood containing 128 cubic feet is a cord. 
But, since wood is usually cut in lengths of 8 feet, it is cus¬ 
tomary to call a pile of wood 8 ft. long, 4 ft. wide, and 4 ft. 
high one cord. One foot in length of this pile is called a 
cord foot and contains 16 cubic feet. 

XII. Table. 

16 Cubic Feet = 1 Cord Foot 
8 Cord Feet = 1 Cord. 

1 Cord = 8 Cord Feet = 128 Cubic Feet. 

Hoiv many cubic feet in 

1. 5 cords ? 2. 7 cords ? 3. 9 cords ? 4. 11 cords ? 

Hoio many cords in a pile of ivood 

5. 4'X4'X88'? 7. 3'X5'X42'? 9 . 8'X8'X24'? 

6. 8'X8'X84'? 8. 5'X7'X40'? 10 . 12'X6'X16'? 


234. 


Circular Measure. 




a. A circle is a plane figure bounded by a curved line, 
called the circumference, every point of which is equally 
distant from a point within, called the center. 







CIRCULAR MEASURE. 


108 


b. An arc is any part of a circumference. 

C. An angle is the opening made by two straight lines 
which meet. It is measured by the size of the opening; the 
length of the sides is immaterial. 

d. Every circle, whether large or small, contains 860 de¬ 
grees. Therefore, an arc of one degree on a large circle is 
longer than an arc of one degree on a small circle, but 
angles of one degree are equal. Compare the arcs of 90° in 
the figure. 

e. An angle is measured by the number of degrees in the 
arc of any circle included between its sides, when the center 
of the circle is the vertex of the angle. 

235. XIII. Table. 

60 Seconds = 1 Minute (') 

60 Minutes = 1 Degree (°) 

360 Degrees = 1 Circumference (cir.). 

1 Cir.=360 Degrees=21600 Minutes=1296000 Seconds ("). 

10 degrees 30 minutes 45 seconds is written 10° 30' 45". 

No confusion will result from having the same signs that are used 
for feet and inches. When necessary to distinguish circular measure 
from time, say “ minutes of arc,” “ seconds of arc.” 

The circumference of the earth at the equator is 24900 
miles; therefore, 1 degree of the equator is 69£ miles long. 
Navigators, for convenience in calculating, call a minute of 
arc one knot. 

60 knots = 69£ miles, or 6 knots = 7 miles (nearly). 

How long is the circumference 

1. If an arc of 1° is 1 foot long ? 

2. If an arc of 1° is 5 feet long ? 

3. If an arc of 5° is 20 feet long ? 

4. If an arc of 90° is 6225 miles long ? (The earth.) 


104 


DENOMINATE NUMBERS. 


5. If an arc of 90° is 1692 miles long? (The Moon). 

6. How many knots round the earth at the equator ? 

7. When a ship is making 18 knots an hour, what is the 
speed in miles ? 

8. A ship goes 6° 20' on the equator in 24 hours. What 
is the speed in knots per hour ? 

9. If the angle in the middle figure is 60° and the cir¬ 
cumference is 8 inches, how long is the arc ? 

10. What is the length in inches of one ten-millionth of a 
quadrant of the earth’s surface ? 

11 . The earth revolves 860° in 24 hours. Through what 
angle does it revolve in 1 hour ? 

12. The circumference of a circle is 8.1416 times its di¬ 
ameter. What is the diameter of a circular race-track 1 mile 
long ? 


236. Time. 

The earth has two motions : One of daily rotation on its 
axis, the other of yearly revolution around the sun. The 
time required for the daily motion has been divided into 24 
equal parts called hours. The time required for the yearly 
motion is not an exact number of days ; it is, in fact, 
365.2422+ days. As it would be inconvenient to count the 
fraction (.2422) of a day, it is omitted from common years. 
But in four years the product amounts to four times .2422, 
which is nearly .97 of a day ; it is, therefore counted as a 
whole day. The years to which this extra day is added are 
called leap years, and are exactly divisible by 4. 

But, again, .97 is .03 less than a whole day, and if we 
consider it a whole day, we will add, in 400 years, 100 
leap days, instead of the correct number, 97. Hence, 
3 years in 400, which would ordinarily be leap years, are 



TIME. 


105 


not counted as such. The years not counted are those 
centuries in which the number of the century is not divis¬ 
ible by 4. Thus, 1700, 1800, and 1900 are not leap years, 
because 17, 18, and 19 are not divisible by 4, but, 2000 will 
be a leap year. 

All years divisible by If are leap years , except centennial years , 
which must be divisible by Jf00. 

Which of the following are leap years ? 

1. 1800. 4. 1888. 7 . 1916. 10 . 2000. 

2. 1824. 5 . 1900. 8. 1925. n. 1776. 

3. 1880. 6. 1902. 9. ,1600. 12 . 1492. 


237. XIV. Table. 

60 Seconds=l Minute (min.) 4 Weeks=l Lunar Month. 

60 Minutes=l Hour (hr.) 865 Days =1 Common Year. 

24 Hours =1 Day (da.) 866 Days =1 Leap Year. 

7 Days =1 Week (wk.) There are 52 Weeks in a Year. 
12 Months =1 Year (yr.) In business, 80 da.=l Month. 
100 Years =1 Century. sec.=Second. mo.=Month. 
865 Days 5 Hours 48 Minutes 46 Seconds=l Solar Year. 
Reduce to integers of loioer denominations : 

1 . .2422 da. 3 . .57 mo. 5 . T 3 T yr. 7 . T 7 ? mo. 

2. .4521 yr. 4 . .19 wk. 6. || yr. 8. T \ mo. 

9 . How many days in 8 score years and 10 ? Add a day 

for every four years. 

10 . A clock which ticks twice in a second, ticks how 
many times in a day ? 

11. How many years does it take for the error in the pres¬ 
ent method of reckoning time to amount to a day ? 

Solution. —In 400 years we add 97 days. We should add only 
.2422X400=96.88 days ; hence in 400 years we add .12 of a day too 
much. 8 % times .12 makes 1. Therefore we will have added 1 day 
too many in 8^X400 years. 


106 


DENOMINATE NUMBERS. 


12 . Our present system was adopted in 1752. What year 
ought to have only 864 days ? . 

13. If the interest on $1 for 1 yr. is 6^, what is the in¬ 
terest for 1 month in cents and for 1 day in mills ? 

14. Reduce 95281 sec. to higher denominations. 

15. Reduce 435.28 da. to higher and lower denominations. 


238. 


XV. Table. 


No. 

Month. Days. 

1. January (Jan.) 31. 

2 . February (Feb.) 28. 

3 . March (Mar.) 31. 

4 . April (Apr.) 30. 

5 . May (May) 31. 

6 . June (June) 30. 


No. 

Month. Days. 

7. July (July) SI. 

8 . August (Aug.) 31. 

9 . September (Sept.) 30. 

10 . October (Oct.) 31. 

11 . November (Nov.) 30. 

12 . December (Dec.) 31. 


Abbreviations are the first three letters of the name, ex¬ 
cept June, July and Sept. 

April is the fourth month, and September, which has 
nine letters, is the ninth month. 

Thirty days hath September, 

April, June, and November. 

The others have 31, except February. February has 28, 
except in leap years, and then 29. 

A day begins at midnight and ends the following mid¬ 
night. Hours before noon are marked a. m., and hours after 
noon are marked p. m. Midnight is written 12 p.m., and 
noon, 12 m. 

It would be better if hours were numbered from 1 to 24. 

p. m. hours would then be their present numbers plus 12. 

1 p. m. w'ould be 13 o’clock ; 6 p.m. 18 o’clock. 


TIME. 


107 


239. 


Difference of Time. 


What is the time from Jan. 15, 1897, to June 8, 1899 ? ' 

Directions.— Write the earlier date un¬ 
der the later, putting number of month 
in place of name. Subtract, counting 30 
days to the month. 

What is the time 

1. From Jan. 8, 1884, to July 7, 1887 ? 

2. From Aug. 27, 1890, to April 5, 1895 ? 

3. From Sept. 18, 1889, to Mar. 16, 1891 ? 

4. From June 6, 1898, to July 4, 1897 ? 

5. From Dec. 1, 1895, to Jan. 1, 1899 ? 

6. The Revolutionary war began April 19, 1775, and 
ended Jan. 20, 1788. How long did it last ? 

7. The first English settlement in the United States was 
at Jamestown, Va., May 28, 1607. How long was that before 
the Declaration of Independence, July 4, 1776? 

8 . Columbus discovered America Oct. 12, 1492. How 
long was it before the first English settlement was made ? 


1899 6 8 

1897 1 15 

2 4 18 


How long did each of the following men live f 

9. Washington. Born Feb. 22, 1732. Died Dec. 14, 1799. 


10 . 

11 . 

12 . 

13. 


Jefferson. 

Grant. 

Lee. 

Webster. 


Born Apr. 2, 1743. 
Born Apr. 27, 1822. 
Born Jan. 19, 1807. 
Born Jan. 18, 1782. 


Died July 4, 1826. 
Died July 23, 1885. 
Died Oct 12, 1870. 
Died Oct, 24, 1852. 

How old were each of the following persons on Oct. 1 , 1896 f 

14. Grover Cleveland. Born Mar. 18, 1837. 

15. Queen Victoria. Born May 24, 1819. 

16. W. E. Gladstone. Born Dec. 29, 1809. 

17. Prince von Bismarck. Born Apr. 1, 1815. 



108 


DENOMINA TE NUMBERS. 


240. Number of Days Between Two Dates. 


1. 

2. 

3. 

4. 

5. . 

6. 

Day of 

7. 

8. 

9. 

10. 1 

11. 

12. 

Jan 

. Feb. 

Mar. 

Apr. 

May. 

June. 

Month 

July. 

Aug. 

Sept. 

Oct. 1 

Nov. 

Dec. 

1 

32 

60 

91 

121 

152 

1 

182 

213 

244 

274 

305 

335 

2 

33 

61 

92 

122 

153 

2 

183 

214 

245 

275 

306 

336 

3 

34 

62 

93 

123 

154 

3 

184 

215 

246 

276 

307 

337 

4 

35 

63 

94 

124 

155 

4 

185 

216 

247 

277 

308 

338 

5 

36 

64 

95 

125 

156 

5 

186 

217 

248 

278 

309 

339 

6 

37 

65 

96 

126 

157 

6 

187 

218 

249 

279 

310 

340 

7 

38 

66 

97 

127 

158 

7 

188 

219 

250 

280 

311 

341 

8 

39 

67 

98 

128 

159 

8 

189 

220 

251 

281 

312 

342 

9 

40 

68 

99 

129 

160 

9 

190 

221 

252 

282 

313 

343 

10 

41 

69 

100 

130 

161 

10 

191 

222 

253 

283 

314 

344 

11 

42 

70 

101 

131 

162 

11 

192 

223 

254 

284 

315 

345 

12 

43 

71 

102 

132 

163 

12 

193 

224 

255 

285 

316 

346 

13 

44 

72 

103 

133 

164 

13 

194 

225 

256 

286 

317 

347 

14 

45 

73 

104 

134 

165 

14 

195 

226 

257 

287 

318 

348 

15 

46 

74 

105 

135 

166 

15 

196 

227 

258 

288 

319 

349 

16 

47 

75 

106 

136 

167 

16 

197 

228 

259 

289 

320 

350 

17 

48 

76 

107 

137 

168 

17 

198 

229 

260 

290 

321 

351 

18 

49 

77 

108 

138 

169 

18 

199 

230 

261 

291 

322 

352 

19 

50 

78 

109 

139 

170 

19 

! 200 

231 

262 

292 

323 

353 

20 

51 

79 

110 

140 

171 

20 

201 

232 

263 

293 

324 

354 

21 

52 

80 

111 

141 

172 

21 

202 

233 

264 

294 

325 

355 

22 

53 

81 

112 

142 

173 

22 

203 

234 

265 

295 

326 

356 

23 

54 

82 

113 

143 

174 

23 

204 

235 

266 

296 

327 

357 

24 

55 

83 

114 

144 

175 

24 

205 

236 

267 

297 

328 

358 

25 

56 

84 

115 

145 

176 

25 

206 

237 

268 

298 

329 

359 

26 

57 

85 

116 

146 

177 

26 

207 

238 

269 

299 

330 

360 

27 

58 

86 

117 

147 

178 

27 

208 

239 

270 

300 

331 

361 

28 

59 

87 

118 

148 

179 

28 

209 

240 

271 

301 

332 

362 

29 


88 

119 

149 

180 

29 

| 210 

241 

272 

302 

333 

363 

30 


89 

120 

150 

181 

30 

211 

242 

273 

303 

334 

,364 

31 


90 


151 


31 

1 212 

243 


304 


365 

Jan 

. Feb. 

Mar. 

Apr., 

May. 

June. 

,Day of 

1 July. 

Aug. 

I Sept. 

Oct. 

Nov. 

Dec. 


I Month! 


In a leap year, Feb. 29th is the 60th day, and dates after 
Feb. 29th are 1 more than the number in the table. 

Jan. 1st is the 1st day, and Dec. 81st the 865th day of the 
year. From this table, the day of the year for any other 
date may be found. 






































TIME. 


109 


What day of the year is July 4th ? 

Look in the column of black figures for 4; in the same row, under 
July, is 185. July 4th is the 185th day of the year. 

What day of the year is 

1 . Feb. 22 ? 2. Apr. 19? 3. Oct. 12? 4. Dec. 25? 

5. Jan. 20? 6. May 28? 7. Apr. 26? 8. Nov. 22 ? 


240 a. 


Exact Difference in Days. 


How many days is it from Washington’s Birthday, Feb. 
22, to the 4th of July ? 

Directions. — Referring to 
the table, July 4th is the 185th 
day ; Feb. 22d is the 53d day; 
the difference is 132 days. Or 
it can be done as on the right. 

In leap years, every date 
after Feb. 29 is one more than 
that shown in the table. 


July 4=185 
Feb. 22 = 58 

Difference =132 


iday, 

Feb. 

Feb. 

6 

Mch. 

31 

Apr. 

30 

May 

31 

June 

30 

July 

4 

132 


What date is 93 days after March 16 ? 

Mar. 16= 75 
93 

June 17=168 


Directions. —March 16th is the 75th day, 
93 days after is the 168th day, which, by the 
table is June 17th. 


Find the exact number of days (in same year) 


1. From Jan. 20 to Apr. 19. 

2. From May 23 to July 4. 

3. From July 4 to Oct. 12. 

4. From Feb. 22 to Dec. 25. 

5. From July 4 to Dec. 25. 
What date is 


11. 

93 

days 

after 

Apr. 

2? 

12. 

63 

days 

after 

Feb. 

22? 

13. 

33 

days 

after 

Jan. 

20? 

14. 

93 

days 

after 

July 

4? 

15. 

63 

days 

after 

Apr. 

19? 


6. From Jan. 19 to July 4. 

7. From Apr. 26 to Sept. 1. 

18. From June 1 to Sept. 1. 
9. From Apr. 19 to July 4. 

10 . From Feb. 22 to Oct. 12. 

16. 123 days after July 4 ? 

17. 33 days after Nov. 22 ? 
98. 67 days before Dec. 25 ? 

19. 85 days before July 4 ? 

20. 93 days before Nov. 5 ? 






110 


DENOMINATE NUMBERS. 


241. 


Longitude and Time. 
Noon 



The meridian of a place is the line drawn from the north 
to the south pole through that place. 

Longitude is the distance in circular measure, or in time, 
east or west of a fixed meridian. The meridian from which 
nearly all nations reckon longitude is that of Greenwich, a 
small place in England, just below London. 

The longitude of a place is east, if the place is east, or 
west, if the place is west, of the standard meridian. It is 
reckoned 180° each way. To reckon it westward 860° would 
be better. 

The earth revolves 860° in 24 hours, therefore it revolves 
15° in 1 hour, 15' in 1 minute, 15" in 1 second. 








LONGITUDE AND TIME. 


Ill 


XVI. 

Table. 

Circular Measure. 

Time. 

15° = 

1 Hour. 

15' = 

1 Minute. 

15" = 

1 Second. 


Since the sun rises and then sets, there must be an instant 
when it ceases to rise and begins to set; at that instant the 
sun is on the meridian and it is noon. 

It is p.m. at all places eastward, and a.m. at all places 
westward, to the meridian of 180°. 


242. Time to Circular Measure. 

It may be observed from the table (Art. 241) that multi¬ 
plying the hours, minutes, and seconds of time by 15 gives 
the corresponding degrees, minutes, and seconds of arc. 

The difference in time between two places being 4 hr. 9 
min. 10 sec., what is the difference in longitude ? 

Hr. min. sec. Analysis . 

4 9 10 Since 1 sec. = 15", 10 sec. = 10X15"=150". 

15 Since 1 min.= 15'. 9 min. = 9X15' =135'. 

00° 135' 150" Since 1 hr. = 15°, 4 hr. = 4X15° =60°. 

62° 17' 30" Ans. 

What is the difference in longitude between two places xohose 


difference in time is 



Hr. min. sec. 


Hr. 

min. 

sec. 

Hr. 

min. 

sec. 

1 . 

2 3 

2? 

6. 

4 

18 

28 ? 

11. 9 

14 

4$? 

2. 

5 3 

4? 

7. 

3 

37 

18? 

12. 4 

59 

18*? 

3. 

0 45 

10? 

8. 

5 

49 

6? 

13. 7 

11 

2*? 

4. 

6 2 

3? 

9. 

1 

56 

52? 

14. 8 

2 

81f? 

5. 

1 11 

0? 

10. 

7 

43 

31 ? 

15. 1 

6 

42$ ? 




112 


DENOMINATE NUMBERS. 


243. Circular Measure to Time. 

It may be observed from the-table (Art. 241) that divid¬ 
ing the degrees, minutes, and seconds of arc by 15 gives the 
corresponding hours, minutes, and seconds of time. 

What is the difference in time between two places when 
their difference in longitude is 81° 17' 80" ? 

a. 15 181° 17' 80" b. 15 181° 17' 80" 

60 120 4 8 

77 150 1 2 

2 hr. 5 mi. 10 sec- 2 hr. 5 mi. 10 sec. 

Analysis. —Since 15°= 1 hr., 31°=as many hours as 15 is contained 
in 31. Similarly for 77' and 150". 

In a we multiply each remainder by 60 and divide the 
product by 15, which is equivalent to multiplying the re¬ 
mainder by 4. The latter method is shown in b. 


What is the difference in time when the difference in longitude is 


1. 18° 

2. 87° 

3. 28° 

4. 74° 

5. 2° 


28' 45"? 
87' 47"? 
59' 0" ? 

0' 24"? 

20 ' 0 " ? 


6. 48° 20'? 

7. 12° 27'? 

8. 30° 19'? 

9. 77° O'? 

10. 116° 26'? 


11. 

12 . 

13. 

14. 

15. 


122 ° 

90° 

71° 

81° 

77° 


23' 19" ? 
3' 28"? 
30"? 
30"? 
36" ? 


3 ' 

40' 

0 ' 


244. i. If a watch, which keeps correct time, gets slow 
when its owner is traveling, is he traveling eastward or 
westward ? 

Arts. Suppose the owner in New York and that his watch shows the 
correct time to be noon. It is past noon at all places to the eastward. 
Now, if the watch were suddenly moved eastward it would register 
noon, when the real time is past noon and, hence, would be slow. 

2. If the watch is 1 hr. slow, how many degrees has he 
traveled ? 

3. Suppose the following possible ! A man leaves New York 
at noon on Friday and travels westward as fast as the sun* 





LONGITUDE AND TIME. 


113 


so that it is noon with, him all the time. Now, it was Fri- 
day noon when he started and Saturday noon when he ar¬ 
rived. Where did it change from Friday noon to Saturday 
noon ? Ans. At the meridian of 180°. 

4. Look on a map having London on it and you will see 
the following statement to be true : Given the longitude of 
two places, if they are both B. or both W., their difference 
of longitude is found by subtraction. Examples : New York 
and Chicago, Paris and Rome. If one is E. and the other is 
W ., it is found by addition. Examples: New' York and 
Paris, Chicago and Rome. 

245. Comparison of Time at Different Places. 

Look at a map of the United States. 

Where does the sun rise ? 

When it is noon at your home, in which direction are 
places where it is past noon ? 

When it is 11 a. m. at Chicago, what is the time on a ship 

75° (5 hours) east of Chicago ? Reckon time on the 24-hour 
system. 

AnALY.is.-The time at Chicago being given, it is later at all places 
east of Chicago. The time on the ship is, therefore, ll-f-5=16 
o’clock=4 p. m. Ans. 

The following table is given for reference : 


City. 
Berlin, 
Chicago, 
London, 
Paris, 
Pekin, 
Rome, 


Longitude. 

13° 23' 45" E. 
87° 37' 47" W. 
0° O' 0". 

2° 20' 0"E. 
116° 26' 0" E. 
12° 27' 0" E. 


City. Longitude. 

Constantinople, 28° 59' 0' 


E. 


New York, 

Rio Janerio, 

SanFrancisco, 

St. Petersburg, 

Washington, 

a. What is their difference in time ? 

b. When it is the time given at the place named , what time is 
it at the other place f 

1. a. New York and Chicago, b. Noon at Chicago. 


74° O' 24" W. 
48° 20' 0"W. 
122° 28' 19" W. 
30° 19' 0" E. 
77° 0' 36" W. 



114 


DENOMINATE NUMBERS. 


2. a. New York and Paris, b. Noon at New York. 

3. a. London and San Francisco, 
b. 8 a. m. at San Francisco. 

4. a. Washington and Constantinople, 
b. 8 p. m. at Constantinople. 

5. a. Washington and San Francisco, 
b. 4 hr. 85 min. p. m. at Washington. 

6. a. St. Petersburg and Pekin. 

b. 2 hr. 14 min. 30 sec. a. m. at Pekin. 

7. a. Berlin and Rome. b. 5 hr. 42 min. p. m. at Rome. 

8. a. Washington and Pekin. 

b. 6 hr. 35 min. p. m. at Pekin. 

9. a. Washington and London. 

b. 9 hr. 26 min. 18 sec. a. m. at Washington. 

10. When it is noon at Washington find the time at every 
other place in the list. 

246. Finding Longitude at Sea. 

Every seagoing vessel carries a chronometer from which 
the correct Greenwich time can be obtained. With an 
instrument called a sextant, the moment when the sun 
ceases to rise in the heavens, which is local noon, can be ob¬ 
served. The Greenwich and local times being known, their 
difference gives the longitude in time. 

The Greenwich time is 10 hr. 16 min. 52 sec. a.m. when 
the sun is on a ship’s meridian. What is the ship’s longi¬ 


tude ? 


Hr. 

12 

10 


min. sec. 


0 0 Ship time. 

16 52 Greenwich time. 


1 43 8 

15 


The longitude is east because it is 
noon at the vessel before it is noon at 
Greenwich. 


25° 47' 0" E. An s. 





DISTANCE AND TIME. 


115 


What is the longitude of a vessel where the navigator observes 
that the sun is on the meridian and notes the correct Greenwich 
time to be 



Hr. 

min. sec. Hr. 

min. 

sec. 


Hr. 

min. 

sec. 

1. 

11 

45 20? e. 16 

42 

15 ? 

11. 

4 

25 

48 ? 

2. 

9 

13 16? 7 . 22 

16 

48 ? 

12. 

19 

23 

52 ? 

3. 

2 

14 51 ? s. 7 

48 

11 ? 

13. 

23 

41 

3 ? 

4. 

13 

1 26 ? 9 . 9 

23 

42 ? 

14. 

7 

18 

17 ? 

5. 

IT 

48 22 ? io. 14 

46 

19 ? 

15. 

10 

2 

6 ? 

247. 

Distance 

AND 

Time. 






Mi 

. per hr. 

Ft. per 

sec. 






1 


1.4667 





A man walks 

3 

== 

4.4 






A horse trots 

7 

= 

10.27 






A bicyclist rides 

10 

= 

14.67 






A steamboat goes 

20 

= 

29.33 






A train goes 

40 


58.67 






A rifle bullet goes 1000 

= 1466.7 





The speed of an object is the distance it goes in a unit of 
time. It is also called the rate. 


< Distance < Distance Distance ^ Distance - Distance 

^i i r n l . n 1 g0e . S .! n . l .. hr ^ Tra '. n . g . 0 . esi . n , l , hr -t Train . goes i in . l . hr ;‘^ Train g° es in 1 hf -|Trai n goes in i hr 

Rate Rate Rate 


Rate 


Rate 


When we say that a pigeon is flying at the rate of 40 
miles an hour, we do not mean that he will actually fly that 
far in that time, because he may alight, but we mean that 
he would fly 40 miles in 1 hour if he continued at the same 
rate as when we observed him. 


NEW YORK 





116 


DENOMINATE NUMBERS. 


In expressing the speeds of objects, the unit of time is 
usually 1 hour or 1 second. The distance is expressed in 
miles or feet, according to the speed. Time, speed, and dis¬ 
tance are so related that when two are given the other may 
be found. 


248. Finding Speed. 

a. What is the speed of a train which goes 12 miles in 36 
minutes ? 

Analysis. —In 36 minutes train goes 12 miles. 

In 1 minute train goes miles. 

In 60 minutes train goes -6-O^i-a miles. 

In 1 hour train goes 20 miles. Ans. 

b. The telegraph posts on a certain railway are 88 yards 
apart. A passenger counts 10 spaces between the posts in 1 
minute. What is the speed of the train ? 

Analysis.— 88 yd.=length of 1 space. 

10X88 yd.=length of 10 spaces. 

=dist. train goes in 1 min. 

60x 10X88 vd.=dist. train goes in 60 min. 

Reducing to miles ^^^^ ^ =30. Ans. 30 miles per hour. 

Find the speed in miles per hour in each of the folloioing 
examples: 

1. A train goes 12 miles in 36 minutes. 

2. A bicyclist rides 4 miles in 20 minutes. 

3. A bird flies 5 miles in 15 minutes. 

4. The wind blows 3 miles in 18 minutes. 

5. A river flows 2 miles in 40 minutes. 

6. The earth moves forward in its orbit 19 miles per sec. 

7. A train goes 3520 yards in 2 minutes. 

8. A chip on a river floats 88 yards in 1 minute. 

9. A very fast steamship goes 528 miles per day. 

10 . A bicyclist has ridden 500 miles in 24 hours. 




DISTANCE AND TIME. 


117 


Eind the speed in feet per second in each of the following : 

11. A boy runs 100 yards in 12 seconds. 

12 . John is 728 yards from me.' I see him strike a bell, 
and 2 seconds after hear the sound. What is the velocity 
of sound ? 

13. Roy rides a bicycle 5 miles in 20 minutes. 

14. A fish swimming at 25 miles an hour. 

15. A train going 60 miles an hour. 

16. Two trains, one 852 feet long, the other 264 feet 
long, pass each other in 7 seconds. One goes at the rate 
of 85 miles an hour. How fast is the other going ? 

17. A ship goes 18 knots per hour. If 6 knots=7 miles, 
what is the speed in feet per minute ? 


249. Finding Time. 

а. From New York to Chicago is 900 miles. How long 
will it take a train at 50 miles an hour to make the trip ? 

NEW YORK. 900 CHICAGO 

* - 1 '- 1 -1—»-J-1-J-1. ■■ -% 

How many fifties in 900? 

At 50 miles an hour , how long will it take a train to make the 
following journeys f 

1* Atlanta 739 Chicago 

• • 

2 . St. Louis 700 New Orleans 

• —---r-• 

3. Washington 654 Atlanta 

• -:---• 

4. Boston 445 Washington 

•--• 

5. New York 217 Boston 

•-• 

б. From New York to Liverpool is 8000 miles. How 
long (days and hours) will it take a steamship, going at the 
rate of 22 miles an hour, to make the trip ? 













118 


DENOMINA TE NUMBERS. 


7. From Baltimore to Hamburg, Germany, is 8900 miles. 
How long will it take a steamship, going at the rate of 21 
miles an hour, to make the trip ? 

8. Sound travels 1092 feet per second. If I see a bullet 
strike a target which is a mile distant, how many seconds be¬ 
fore I will hear the sound ? Sight is practically instantaneous 
for short distances. 

9. A cannon-ball goes 2000 feet per second. How long 
will it take to go 5 miles ? 

10 . If the Mississippi flows 2-J miles per hour, how long 
(in highest denominations) does it take for water to flow from 
its source, Lake Itasca, to its mouth, a distance of 8162 
miles ? 

11. The sun is 92,500,000 miles from the earth. Light 
travels 188,000 miles per second. How long (min. and sec.) 
is taken by the sun’s light in reaching the earth ? 

12. The longest tunnel in the world is St. Gothard, 9| 
miles long. What time is taken by a train in going through 
it, traveling 22 feet per second? 

13. The entire length of the suspension bridge between 
New York city and Brooklyn is 5980 feet. How long (min.) 
will a cab be in crossing it at the rate of 4 miles per hour ? 

14. How long (in highest denominations) would a train, at 60 
miles per hour, traveling night and day without stopping, 
take in going to the sun, 92,500,000 miles ? 

15. A baseball player can run 100 yd. in 12 sec. How 
long would it take him to run the bases; they are 90 ft. apart ? 

250. Miscellaneous Examples. 

The pupil is advised to draw a diagram whenever it is possible. A 
diagram usually makes the solution easier. 

1. Two men travel in the same direction. A is 50 miles 
ahead of B; but B travels 27 miles a day and A 28. In how 
many days will B overtake A ? 



DISTANCE AND TIME. 


119 


2. If a dog runs 7 feet while a rabbit runs 5, how far will 
the rabbit run while the dog run 3 56 feet ? 

3. A can walk 15 miles in 4 hours and B can walk it in 
5 hours. If they start at the same time and walk towards 
each other from places 50 miles apart, how far from each 
other will they be in 1 hour? How long before they will 
meet? How far from A’s starting point? (Draw diagram.) 

4. A boy rode a bicycle 125 miles. The first half of the 
distance he rode at the rate of 12§ miles per hour; the last 
half at 10^ miles per hour. How many hours did it take 
him ? 

5. A leaves Atlanta at a certain time and rides a wheel 
at the rate of 12^ miles an hour. 2f hours after A started, B 
sets out to overtake him and rides at the rate of 15f miles 
an hour. How far apart are A and B 4 hours after B sets 
out ? (Draw diagram.) 

a. A man goes f of a mile in f- of an hour. How long would 
he take in going f of a mile ? 

Let £=the time it takes to go one mile. 

§.r=f of an hour. Statement. 
i.r= T 3 ^ of an hour. i x =^ of an hour. 
x== t 9 o of an hour. f:r=f ^ of an hour. Ans. 

6. A walks j of a mile in 15 minutes. How long will it 
take him to walk 1 mile ? 

7. B walks f of a mile in 9 minutes. How long will it take 
him to walk 1 mile ? 

8. C walks | of a mile in of an hour. How long will it 
take him to walk 1 mile ? 

9. Mr. Man goes f of a mile in f of an hour. How long 
will he take in going f of a mile ? 

10 . A horse trots $ of a mile in 1 minute. How long will 
he take to trot § of a mile ? 


120 


DENOMINA TE NUMBERS. 


11. A bicyclist rides 18f miles in 1 hour. How long will 
he take to ride 187 miles ? 

12. A train goes f of a mile in 1£ minutes. How long would 
it be in going f of a mile ? 

13. A man walks 18| miles in 4f hours. How far could he 
go in 9f hours ? 

14. A horse travels 6f miles in 1| hours. How far can he 
go in 5f hours ? 

15. A freight-train 2200 feet long passes a man in 50 
seconds. What is the rate of train in miles per hour? 

16. The minute and hour hands of a city clock are the 
same length. The end of the minute hand travels 5 feet in 
1 hour. How far does the end of the hour hand go in the 
same time ? 

W. , E. C. 

17. I_ 20 yd- _j_i Walter is 20 yards from 

Ella and is walking 12 times as fast as she. How far must 
he walk before overtaking her ? 

The Clock* Problem.— At 4 o’clock 
the hands are as shown. During the 
next hour, the minute hand moves 60 
minute spaces while the hour hand 
moves 5. The minute hand, therefore, 
travels 12 times as fast as the hour hand. 

As the minute hand moves forward a 
certain distance, the hour hand moves 
forward yV of that distance. Hence, yi 
of the distance the minute hand moves 
is the gain it makes on the hour hand. 

b. At what time between 4 and 5 o’clock are the hands of 
a clock together ? 

Analysis. —The minute hand is 20 minute spaces behind the hour 
hand and must gain this much on the hour hand before the two hands 
are together. But this gain is ^ of the whole distance the minute 
hand must move. Hence, 








DISTANCE AND TIME. 


121 


H of the number of minute spaces the minute hand must move=20. 
tV of the number of minute spaces the minute hand must move=f$. 
tI of the number of minute spaces the minute hand must move=21j 9 x. 

The time when the hands are together is, therefore, 21 T 9 T min¬ 
utes after four. 

When are the hands of a clock together between 

18. 3 and 4 o’clock ? 20. 5 and 6 o’clock ? 

19. 9 and 10 o’clock ? 21. 7 and 8 o’clock ? 

When are the hands of a clock opposite each other between 

22. 8 and 9 o’clock ? 23 . 11 and 12 o’clock ? 

When are the hands fifteen minutes apart (tw r o answers) between 

24. 6 and 7 o’clock ? 25. 4 and 5 o’clock ? 

26. A clock which loses 5 minutes a day is set correctly 

at 12 noon Monday. What will be the correct time when 
the clock indicates noon the next Monday ? Note 23f£ clock 
hours=24 true hours. 

27. A horse has trotted a mile in 2 min. 4 sec., and a bi¬ 
cyclist has ridden a mile in 1 min. 56 sec. If the horse is 1 
mile ahead and both going at the above rates, how far must 
the bicyclist ride before overtaking the horse ? 

28. Telegraph poles are 66 yards apart. A passenger on a 
train counts 112 spaces between the poles in 4 minutes. 
What is the rate of the train in miles per hour ? 

29. A circular track is two miles round. Two bicyclists 
start from the same point in opposite directions—one at the 
rate of 12, the other 15 miles per hour. How far will the 
faster have gone when they meet ? 

30. Two trains start at the same time towards each other, 
one from Washington at 60 miles an hour, the other from Bal¬ 
timore at 40 miles an hour. When they meet, the faster 
train has gone 8 miles further than the other. How far is 
it from Washington to Baltimore ? 


122 


DENOMINATE NUMBERS. 


251. Work and Time. 

What is work ? 

When a man carries brick, hoes corn, or rides a bicycle, 
he does work. When steam pushes an engine or turns wheels 
in a factory, steam does work. When electricity pulls the 
street-car or clicks the telegraph instrument, it does work. 
When water turns a mill-wheel, or a weight runs a clock, 
the falling body does work. When a horse draws a wagon 
qr carries a man, he does work. 

A force does work when it overcomes resistance and produces 
motion. 

A desk holding up books does no work; likewise, a boy 
holding up a load does no work, however tired he may get. 
Work is never done except when there is motion. 

T he unit of work is the work done in lifting one pound through 
a height of one foot. It is called a foot-pound. 

If 2 pounds are lifted 8 feet, 6 foot-pounds of work are 
done. 

252. How much work is done in lifting 

1. 8 pounds 5 feet ? 4. 1492 pounds 85 feet ? 

2. 4 pounds 7 feet ? 5. 5280 pounds 75 feet ? 

3. 7 pounds 9 feet ? 6. 1760 pounds 98 feet ? 

7. Brick-carriers ascend ladders with a load of 90 pounds. 

How much work will a carrier do in an hour if he takes 25 
loads up 60 feet high ? 

8. A man’s labor per day is equivalent to lifting 800 tons 
1 foot. To how many foot-pounds is this equal ? 

9. A woman weighing 100 pounds walks up stairs 15 times 
during the day. There are 15 steps, each 9 inches high. 
How much energy does she spend in going up during the day ? 

10. How much energy is given out by 100 cubic feet of wa¬ 
ter falling 12 feet ? 1 cubic foot water weighs 62£ pounds. 


WORK AND TIME. 


128 


253. No direct reference to time is necessary when 
quantity of work is considered. If we want to know how 
much work a laborer has done, we look at the pile of dirt he 
has moved, or the number of bricks he has carried, and the 
total amount of work done is the same whether he spent 
hours or days in doing it. But with practical men, “time 
is money,” and the laborer who can do the most work in the 
least time is the one preferred. Laborers, therefore, can 
hire themselves to do a given amount of work, or to work a 
given length of time—to work by the piece or by the day. 

a. A can do a piece of work in 8 days; B could do the same 
work in 4 days. How long would both be in doing it ? 

Analysis.— A can do % of it in 1 day. 

B can do 14 °f it i n 1 day. 

Together they would do of it in 1 day 

=t\ in 1 day, 
or yV in 7 of a day. 

All of it, if? in V" days. V' days =lf days. Ans. 

1. A can do a piece of work in 2 days ; B could do the 
same work in 4 days. How much would each do in 1 day ? 
How long would both be in doing it ? 

2. A could do a piece of work in 8 days ; B could do it in 
the same time. Hotv long would both take to do it ? 

3. If 6 men do a job in 5 days, how long would 4 men 
take ? 

4 . One faucet empties a cask in 9 minutes; another 
empties it in 6 minutes. What part of the contents would 
they empty in 1 minute ? How long would it take both to 
empty it ? 

5. 6 men could do a job in 5 days. If 2 men quit when 
the work is half finished, how long would it take for the 
other 4 men to complete it ? 


124 


DENOMINATE NUMBERS. 


6. Two faucets can empty a cask—one in 6, the other in 9 
minutes. The first has been open 2 minutes, when the 
second is opened. How much longer before the cask will be 
empty ? 

Note.— One horse-power is equal to 33000 foot-pounds per minute. 

7. Brick-carriers ascend ladders with loads of 90 pounds 1 
foot per second. How many such men equal 1 horse-power ? 

8. Niagara Falls is 175 feet high. 1 ton of water will fall 
that far in 8^ seconds. What horse-power would it produce ? 

9. W hat is the horse-power of a dredging machine which 
lifts 180 tons of earth 10 feet high per hour ? 

10 . What is the horse-power of a grain elevating engine 
which lifts 550 bushels of wheat 75 feet high in 9 minutes ? 
A bushel of wheat weighs 60 pounds. 


254. Measurement of Temperature. 

The measure of temperature, like the measure of every 
other quantity, is determined by a comparison of the given 
quantity to a fixed quantity of like kind. 

The temperatures of melting ice and of boiling water, un¬ 
der certain conditions, do not vary, and are, therefore, se¬ 
lected as the standards of comparison. 

Three systems of marking a thermometer are employed_ 

Fahrenheit, Reaumur, and Centigrade. A thermometer is 
graduated by putting it in melting ice and the place to 
which the mercury falls is marked. It is then put in steam 
over boiling water and the place to which the mercury rises 
is marked. 

Mr. Fahrenheit marked the temperature of melting ice 82°, 
and that of boiling water 212°. 

Mr. Reaumur marked the temperature of melting ice 0°, 
and that of boiling water 80°. 



TEMPERATURE. 


125 


The Centigrade thermometer (of the metric system) marks 
the temperature of melting ice 0°, and that of boiling water 
100 °. 

Therefore, between the temperatures of melting ice and 
boiling water there are 180 Fahrenheit degrees, 80 Reaumur 
degrees, and 100 Centigrade degrees. 

F.= Fahrenheit. R.= Reaumur. C.= Centigrade. 


255. f. 

212-r 


R. 


77 -- 


C. 

\ 00 -r Boiling 


F. R. C. 

180° = 80° = 100° 
9 ° = 4° - 5° 


18° = 

8 ° = 

10 ° 

27° = 

12 ° = 

15° 

86 ° = 

16° = 

20 ° 

45° = 

20 ° = 

25° 

54° = 

0 

80° 

etc. 

etc. 

etc. 


■Freezing 


a. 


What is 122° F. on C. scale? 
9 ° F.= 5° C. 

1°F.= f° C. 

90° F.= 50° C. Ans. 


b. 90° F. on JR. scale? 

9 o F _ 40 R 

1° F.= f° R. 

90° F.= 40° R. Ans. 


c. 


What is 60° C. on F. scale f 


5° C.= 
1° C.= 


9° F. 
t°F. 


60° C.= 108° F * 


d. On R. scale f 
50 C.= 4° R. 

1 ° C.= f° R. 

60° C.= 48° R. Ans. 


*108° F. is the number of degrees above freezing point. But 
freezing point being marked 82°, 108° above this should be 
marked 82°+108°== 140° F. Ans. 





126 


DENOMINATE NUMBERS. 


What is 

1. a. 0° C. on Fahrenheit scale ? 

2. a. 68 ° F. on Centigrade scale ? 

3. a. 77° F. on Centigrade scale ? 

4. a. 56° F. on Centigrade scale ? 

5. a. 28° R. on Centigrade scale ? 

6. a. 82° R. on Centigrade scale ? 

7. a. 45° R. on Reaumur scale ? 

8. a. 60° C. on Reaumur scale ? 

9. a. 75° C. on Reaumur scale ? 
10. a. 4° C. on Reaumur scale ? 


b. On Reaumur scale ? 
b. On Reaumur scale ? 
b. On Reaumur scale ? 
b. On Fahrenheit scale ? 
b. On Fahrenheit scale ? 
b. On Fahrenheit scale ? 
b» On Fahrenheit scale ? 
b. On Fahrenheit scale ? 
b. On Fahrenheit scale ? 
b. On Fahrenheit scale ? 


256. Miscellaneous Problems. 

1. The deck of Noah’s ark was 800 cubits long and 50 
cubits wide. If a cubit is 20 inches, what was the area of 
the deck in square feet ? 

2. A screw advances fV of an inch at each turn. How 
many turns must be made to advance it 4J inches ? 

3. What is the weight'of a bucket of water, if the bucket 

holds a peck ? 1 cubic foot of water weighs 1000 ounces. 

4. How many telegraph poles 88 yards apart are there in 
a mile ? 

5. A standard-gauge railway is 4 feet 8| inches wide. 
What length of track encloses an acre between the rails ? 

6. An acre has 12,000 cotton plants on it. It averages 5 
bolls to the plant and half an ounce of seed and cotton to 
the boll. If £ by weight is lint cotton, how many pounds of 
lint are there on the acre ? 

7. The area of the United States is 8,501,410 square 
miles and the population in 1890 was 62,622,250. If equally 
divided, how many acres would each person have owned ? 



MISCELLANEOUS PROBLEMS. 


127 


8. The distance around the earth is 24,900 miles. How 
many miles in a degree of latitude ? 

9. A postage stamp is J inches long and f inches wide. 
How many will cover an envelope 5^ inches square ? 

10 . How many gallons, during a rain of 1 inch, fall on an 
acre of ground ? 

11. If a house rents for $300 a year, how long would it 
rent for $2295 ? 

12. How many tons of water, during a rain of 1 inch, fall 
on a square mile ? 

13. One cubic inch of distilled water at 62° F. weighs 
252.458 grains. What is the weight in ounces of 1 cubic foot? 

14. When heated from the freezing to the boiling point, 
water expands fa of its volume. A cubic foot of water at freez¬ 
ing point weighs 62| lbs. What is its weight at boiling point ? 

15. When ice melts, it contracts fa of its volume. A 
cubic foot of water at freezing point weighs 62^ pounds. What 
is the weight of a cubic foot of ice ? 

16. The atmosphere presses 14.7 pounds per square inch. 
What is the pressure in tons on a boy, the surface of whose 
body is 2000 square inches ? 

17. The tropics of Capricorn and Cancer are each 23° 30' 
from the equator. How far apart are they in miles ? 
1°=69£ miles. 

18. How many miles is it from the arctic circle to the 
north pole ? The distance is 23° 30'. 

19. The highest latitude reached before 1896 was by 
Messrs. Lockwood and Brainerd, who went to latitude 83° 24'. 
How many miles is that from the north pole ? 

20. Mount Everest is 29002 feet high. Reduce this to miles. 

21. The great Napoleon I. was born Aug. 15, 1769, after 
Christ. We believe he is still living. How old will he be 
at the anniversary of the death of his body, May 5, 1900 ? 


128 


DENOMINATE NUMBERS. 


22. The greatest ocean depth yet measured is 29180 feet, off 
the northeast coast of Japan. Reduce this depth to miles. 

23. The blood in the human body weighs 84 pounds. If 6 
ounces of blood pass through the heart at each beat and 
the heart beats 80 times per minute, how many seconds does 
it take for all the blood in the body to pass through the 
heart ? 

24. A hen agrees to lay a great gross of eggs at the rate of 
1 a day. If she lays the first egg on July 4, 1900, on what 
day will she lay the last egg ? 

25. As we go down mines into the earth the temperature 
increases 1° F. for every 50 feet of descent. How deep is it 
where water willlooil ? Water boils at 282° F. at this depth. 

Take 60° as temperature of water at sea-level. 

26. At the sea-level water boils at 212° F., and the boiling 
temperature falls 1° F., for every 558 feet of ascent. How 
high is a mountain on which wat'er boils at 71° C.? 


257. REVIEW. 

Reduce 

1 . 8 wk. 5 da. 18 hr. 47 min. to min. 

2. 5 A. 150 sq. rd. 25 sq. yd. 5 sq. ft. to sq. ft. 

3. 10000 seconds to higher denominations. 

4. 10000 sq. ft. to higher denominations. 

5 . | wk. to integers of lower denominations. 

6. t 9 t mi. to integers of lower denominations. 

7. .2758 sq. mi. to integers of lower denominations. 

8. 7 A. 90 sq. rd. 25 sq. yd. to a fraction of a sq. mi. 

9 . 9 cwt. 75 lb. 12 oz. to a decimal of a cwt. 

10 . What part of 2 sq. mi. 100 A. is 500 A. 25 sq. rd.? 

11. Reduce to integers of lower denominations, then add, 
5 rd., 16 mi., f yd., f rd. 

12. From 8 A. 9 sq. yd. take 1 A. 150 sq. rd. 8 sq. ft. 



REVIEW. 


129 


13. If the tuition for each pupil in a London school is £7 
9s. 6d. per year, what is the income in English money of a 
teacher who has BO pupils ? 

14. From Baltimore to Washington is 40 miles. How 
many telegraph poles 80 yards apart are required for that 
distance ? 

15. What is the cost of rails per mile for a railroad using 
60 lb. (per foot) steel rails @1.1/ per lb.? 

16. If quinine is bought at $4 alb. Avoir, and sold at 50/ 
an oz. Apoth., what is the gain on 100 lb. Avoir.? 

17. A bale of paper weighs 180 lb. What is the weight 
of a book containing 240 pages, if each sheet in the bale 
makes 8 pages of the book ? 

18. What is the area in A. of a rectangular field 210'X420'? 

19. How many gallons will a bathtub 5'X2'X2' hold ? 

20. A bushel of oats weighs 82 lb. How many bu. will a 
bin 4'X4'X8' hold? 

21. Will 1900 A. D. be a leap year? 

22 . What was the exact age of Lincoln at his death ? 
Born Feb. 12, 1809, died Apr. 15, 1365. 

23. What is the exact number of days in a leap year from 
Feb. 29 to July 4? 

24. What is the exact number of days from July 4, 1896, 
to Feb. 22, 1897? ‘ 

25. What is the difference in time between Washington 
and Chicago ? 

26. What is the time in New York when it is 10 hr. 80 min. 
at St. Louis ? Their difference in time is 1 hr. 5 min. 1 sec. 
What is the longitude of St. Louis ? 

27. If the temperature of your blood is 98° F, what is it 
on a Centigrade thermometer ? 

28. The temperature of a schoolroom should be 20° C. 
What should it be on a F. thermometer ? 


180 


DENOMINATE NUMBERS. 


258. The preceding pages tell of the measures in most 
common use in the United States. In different localities, 
however, custom permits the use of certain other measures 
which need not be learned from arithmetic. 

It is to be regretted that our memories must be taxed 
with such an arbitrary system of weighing and measuring. 
It is very unfortunate that we have three scales for weigh¬ 
ing, three for measuring lengths, and two for measuring 
volumes. It is to be hoped, too, that the time will soon 
come when we shall have an international currency. 

In 1789 a commission of scientists was appointed by the 
French government for the purpose of remodeling the French 
system of weights and measures. The result of their labors 
was a new and excellent systein, combining simplicity with 
utility. It is called the Metric System, and is now used by 
all scientists and nearly all civilized nations. It was legal¬ 
ized in this country in 1866. 

It is hoped that every pupil of this book will become so 
familiar with this system that he will see its many ad¬ 
vantages and exert his influence for its practical adoption. 


Miscellaneous Measures. 


4 Inches — 1 Hand. 

9 Inches — 1 Span. 

120 Fathoms = 1 Cable. 

6080 Feet = 1 Knot. 


2^ Feet — 1 Military pace. 
31^Gallons= 1 Barrel. 

2 Barrels = 1 Hogshead. 

100 Sq. ft. = 1 Sq. of flooring. 


3 Knots 


= 1 League. 4 Roods = 1 Acre. 


1 Chain 
100 Centimes 


= 4 Rods = 100 Links. 
== 1 Franc = 19.3/. 



THE METRIC SYSTEM. 


1B1 


259. THE METRIC SYSTEM. 



One Decimeter. 


A=4 inches. B=10 centimeters=100 millimeters. 

a. The first and most important feature of the metric sys¬ 
tem is that every table of measures is written on the scale of 
10. This gives it all the advantages of decimal notation. 

b. The second feature is that the same prefixes for multi¬ 
ples and subdivisions are used in all the tables. 

C. The third feature is the name and quantity of each 
unit. 


260. 

Prefixes. 



Multiples. 

Subdivisions. 


Greek. 

Meaning. 

Latin. Meaning. 

Myria 

M. 10000 

deei d. 

.1 

Kilo 

K. 1000 

centi c. 

.01 

Hekto 

H. 100 

milli m. 

.001 

Deka 

D. 10 



10 

10 10 

10 10 

10 

Myria- Kilo- 

Hekto- Deka- 

deci- centi- 

milli 

261. 

Units. 




Metric is derived from the Greek word metron , which 
means a measure . 

Length. —The unit of length , the unit from which all the 
others are derived, is called a meter (also derived from 
metron). A meter is the length of a certain bar of metal 
kept at Paris. It was intended to be, and is nearly, one 
ten-millionth of the distance from the equator to either pole. 

1 meter=39.37 inches. 












132 


THE METRIC SYSTEM. 


Weight. —The unit of weight is called a gram. 

It is the weight of one cubic centimeter of water 
at its maximum density. The figure is exact size. 

1 gram=15.4 grains. 1000 grams=2.2 pounds. 

Volume. — The unit of volume is called a litGP (pro¬ 
nounced lee-ter). It is the volume of a cubic decimeter. 

1 liter=l liquid quart=.9 dry quart. 

The metric tables for length, weight, and volume are 
formed by placing the prefixes before these three units. 

Abbreviations for the prefixes and units are the initial 
letters of each. The subdivisions and units begin with a 
small letter, the multiples with a capital. 



262. I. Length. 

Myria-meter, Mm. 
Kilo -meter, Km. 
Hekto-meter, Hm. 
Deka -meter, Dm. 

meter, m. 
deci -meter, dm. 
centi -meter, cm. 
milli -meter, mm. 


II. Weight. III. Volume. 

-gram, Mg. -liter, Ml. = 10000 units, 
-gram, Kg’, -liter, Kl. = 1000 units, 
-gram, Hg. -liter, HI. = 100 units, 
-gram, Dg. -liter, Dl. = 10 units. 

gram, g. liter, 1. = Unit. 

-gram, dg. -liter, dl. = .1 of unit, 
-gram, eg. -liter, cl. — .01 of unit, 
-gram, mg. -liter, ml. == .001 of unit. 


263. A 5/ nickel weighs 5 grams and is 2 centimeters in 
diameter. A liter is one liquid quart. A liter of water 
weighs one Kilogram. 1 Kilogram=2.2 pounds. 


264. NOTATION. 

In United States measure, long distances are usually ex¬ 
pressed in miles, short ones in feet; heavy weights in tons, 
light ones in pounds; large volumes in bushels, small ones in 
quarts; although the other denominations are employed when 








NOTATION. 


133 


desirable. In the metric system, also, certain denominations 
are usually employed to express large and others to express 
small quantities, though the given quantity might be ex¬ 
pressed by any denomination. The abbreviations of the 
ones usually employed are in black type. 


265. Used in measuring same quantities. (For reference.) 

1 m.=5=1.1 yd. meter and yard. lyd.=.914m. 

1 Km.=.621 mi. Kilometer and mile. l mi.=1.61 Km. 

11.=1.06 qt. liq. liter and quart. 1 qt.=.946 1. 

1 HI.=2.84 bu. Hektoliter and bushel. 1 bu. = .352Hl. 

1 g.=.035 oz. Av. gram and ounce. 1 oz.=28.38 g. 

1 Kg.=2.2 lb. Kilogram and pound, l lb. = .454 Kg. 

The metric ton (=1000 Kg.) is used in weighing coal, 
iron, hay, and other heavy weights. 

266. What is the rule for abbreviations ? 

11. deciliter. 

12. Dekagram. 

13. Kilometer. 

14. Myriagram. 

15. Dekaliter. 


Abbreviate : 

1. meter. 

2. gram. 

3. liter. 

4 . centimeter. 

5. Hektogram. 


6. millimeter. 

7. Hektoliter. 

8. Kilogram. 

9. centimeter. 
10 . milligram. 


267. 

Read : 





a. 

b. 

C. 

d. 

e. 

f. 

l. 7 m. 

3 mg. 

7 HI. 

5Dg. 

4 cm. 

9 Kg. 

2. 9 g. 

9 mm. 

4Dg. 

9 dm. 

8 eg. 

7 dm. 

3. 5 1. 

6 Kg. 

8 Km. 

3 Mm. 

6 Dl. 

6 HI. 

4. 8 Kg. 

5 HI. 

9 eg. 

7 HI. 

9 Mg. 

5 eg. 

5. 6 HI. 

2 Kg. 

3 dm. 

2 cm. 

8 Kg. 

8 mg. 

268. 

Metric tables are written on the decimal scale—that 


is, ten units of any order make one unit of the next higher 
order (Art. 206). 


184 


THE METRIC SYSTEM. 


5 hundreds 2 tens 8 units=528 units. 

5 dollars 2 dimes 8 cents=528 cents. 

5 Hektoliters 2 Dekaliters 8 liters=528 liters. 

269. In the metric system, as in our money system, any 

denomination may be used as the unit. 

$5.28=52.8 dimes=528 cents. 

5.28 HI.=52.8 Dl. =528 liters. 

270. The denomination of a denominate number written on 
the decimal scale is the name of the units place. 

$25.67=2 eagles 5 dollars 6 dimes 7 cents. 

84.81 HI.=8 KL 4 HI. 8 Dl. 1 1. 

In 29.45 dm., 9 is dm. In 125.87 Kg., 5 is Kg. 

271. It is important to remember that the separatrix is 
always on the right of units place. 


272. NUMERATION. 

Myria-, Kilo-, Hekto-, Deka-, gram, deci-, centi-, milli-. 

Memorize the prefixes and unit in their proper order from 
left to right and right to left. This is the key to the system. 

a. Enumerate 2.2046 Kg. 

Directions. —Beginning at units place , then successively to the 
right, name the orders : “Kg., Hg., Dg., g., dg.” Read the number : 
“2 and 2046 ten-thousandths Kilograms.” 

Myria-, Kilo-, Hekto-, Deka-, meter, deci-, centi-, milli-. 

b. Enumerate 8.987 m. 

Directions. —Beginning at units place , then successively to the 
left, name the orders : “m., Dm., Hm., Km.” Read the number: 
“3937 meters.” 

273. In enumerating, always begin at units place. 

The quantity might be read by telling the number in each 
order, but this is not the practical method. 




REDUCTION. 


135 

Enumerate: 




1 . 2.2046 Kg. 

6 . .94625 Kl. 

11 . 

3937 m. 

2 . 1.6098 Km. 

7. .3937 Km- 

12 . 

3524 1. 

3. 1.101 1. 

8. .45359 Kg. 

13. 

15432 g. 

4. 15.432 g. 

9. .06479 Mg. 

14. 

393704 mm. 

5 . 1.094 m. 

10 . 1.60932 Mm. 

15. 

154323 mg. 

274. 

REDUCTION. 




Descending. —Reduce 3.78|> Hektoliters to liters. 
Directions —Begin at units place and enumerate to the right : 
“Hekto, Deka, liter.” Place the separatrix on the right of the required 
denomination. Ans. 378.5 1. 

Ascending. —Reduce 8987043 centimeters to Kilometers. 
Directions. —Begin at units place and enumerate to the left : 
“centi, deci, meter, Deka, Hekto, Kilo.” Place the separatrix on the 
right of the required denomination. Ans. 39.37043 Km. 

Annex or prefix zeros when necessary. 


Reduce: 


1 . 

3.785 HI. to 1. 

15. 

39.3704 Hm. to dm. 

2. 

1.6093 Km. to m. 

16. 

22.046 Hg. to dg. 

3. 

2.2046 Kg. to g. 

17. 

378.5 HI. to 1. 

4. 

3.937 Dm. to cm. 

18. 

39.37 Km. to m. 

5. 

2.2046 Hg. to dg. 

19. 

22.04 Kg. to g. 

6. 

35.24 HI. to 1. 

20. 

3.937 Mm. to m. 

7. 

.3937 Dm. to dm. 

21. 

3937043 cm. to Km 

8. 

.4536 Kg. to eg. 

22. 

22046 g. to Kg. 

9. 

.02838 Kl. to dl. 

23. 

37850 1. to Kl. 

10. 

.22046 Dg. to mg. 

24. 

393704 m. to Km. 

11. 

3.785 Kl. to cl. 

25. 

22046 eg. to Hg. 

12. 

3.93704 Mm. to m. 

26. 

37850 cl. to HI. 

13. 

2.2046 Kg. to dg. 

27. 

393704 cm. to Hm. 

14. 

1.10125 Dl. to cl. 

28. 

22046 mg. to g. 



186 


THE METRIC SYSTEM. 


29. 87850 ml. to 1. 

30. 89.37 mm. to m. 

31. 22.04 eg. to Hg. 

32. 37.85 cl. to HI. 

33. 39.37 cm. to Hm 

34. 22.04 g. to Kg. 


35. 37.85 1. to Kl. 

36. 393.7 cm. to Km. 

37. 220.4 eg. to Hg. 

38. 378.5 cl. to HI. 

39. 3.937 mm. to dm 

40. 2.204 mg. to dg. 


275. FUNDAMENTAL OPERATIONS. 


Addition, Subtraction, Multiplication, and Division are 
performed as with ordinary numbers. 

Since only like numbers can be added or subtracted, the 
given numbers must, if necessary, be reduced to the same 
denomination. Reduce distances to Km., m., cm., or mm.; 
weights to Kg., or g.; volumes to HI., or 1. (Art. 264). 


Add: 

Meters. 

a. .393704 Hm. = 39.3704 
621.4 mm. — .6214 

254 cm. = 2.54 
3.048 dm. = .3048 

.0016 Km. *= 1.6 

44.4366 

Add : 


Add: 

Liters. 

b. 3.785 1. = 3.785 
35.24 cl. = .3524 

.9463 Kl. =946.3 
1.101 Dl.= 11.01 
.264 HI. =s 26.4 


1. 15.43 g., 2.204 Kg., .06479 Kg., and 26792 mg. 

2. 39.3704 m., .62137 Km., 254 cm., and 1.6093 Km. 

3. .90816 HI., .02838 Kl., 85.24 cl., and 3785 ml. 

4. 154323 mg., 2.2046 Kg., .0648 Kg., and 4536 eg. 

5. .0393704 Km., .621 Km., 1.609 Hm., and 1094 m. 

6. .90816 HI., 1.056 Dl., .02838 Kl., and .264 1. 

7. 2.2046 Kg., 1543 eg., .00648 Mg., and 453592 mg. 

8. 39370.4 mm., .062138 Mm., 2.54 m., and 3048 cm. 

9. 1.60938 Km., 10986 cm., 6213.8 Hm., and .305 Hm 

10. 2.2046 Kg., 15432 mg., .0648 Kg., and .4536 Hg. 





FUNDAMENTAL OPERATIONS. 


137 


276. From 

1 . 39.8704 Km. subtract 62137677 mm. 

2. 3785 1. subtract 3.524 Kl. 

3. 2.2046 Kg. subtract .064799 Mg. 

4. 1.6093 cm. subtract 1.0936 mm. 

5. .90816 Dl. subtract 1.056 1. 

6. .3732 Hg. subtract 22046 mg. 

7. 393704 mm. subtract .91439 Hm. 

8. 8524 HI. subtract 90.816 Kl. 

9 . 2204.6 Kg. subtract 154.3234 Mg. 

10. 3937.04 cm. subtract 10936 mm. 


277. a. Multiply and b. Divide (Answer in m. 
3 decimals.) 


g- 


or 1., to 


1. 39.3704 m. by 4. 

2. .62138 Km. by 6. 

3. 2.54 cm. by 8. 

4. .3048 Dm. by 7. 

5. .91439 Mm. by 9. 

6. 3.785 1. by 3. 

7 . 35.24 cl. by 6. 

8. 94625 Kl. by 8. 

9. 11012 HI. by 9. 

10. 26.4 Kl. by 4. 

21. 1 lb. Avoir.=.453593 Kg. 


11 . 2204.6 g. by 3. 

12. 1543.2 eg. by 5. 

13. .0648 Kg. by 7. 

14. 4536 mg. by 9. 

15. 37.3 Kg. by 4. 

16. 16093 mm. by 6. 

17. .02838 Ml. by,8. 

18. 2679.2 eg. by 6. 

19. 3.2808 Km. by 7. 

20. 1056 ml. by 8. 

How many Kg. in 2.2 lb.? 
How many meters 


22. A train goes 680 Km. in 9 hours, 
does it go in 1 minute ? 

23. 1 mile = 1.609 Km. It is 24900 miles around the 
earth. How many Km. is it ? 

24. A circle is 1 Km. around. What is its diametei in 

meters ? Divide 1000 m. by 3.1416. 

25. The atmosphere presses 6.68 Kg. per sq. in. 
the pressure in Kg. on 1 sq. m.? 


What is 


188 


THE METRIC SYSTEM. 


278 . 


Square Measure. 

Centimeters 


1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

IO 

2 










3 










4 



Each side of this figure is 1 dm. or 
10 cm. Its area is 1 sq. dm. or 100 
sq. cm 

If four equal sides were put to this, 
its volume would be a liter. A liter 
of water weighs 1 kilogram. 




S 






6 






>7 






8 


--1 

1 

1 

1 








9 

> 

1 

1 

1 

1 








io 


I 

1 

1 ✓ 

1 











Inches 




IV. 

Table. 



100 

Sq. 

mm. — 

1 

Sq. 

cm. 

(cm. 2 ) 

100 

Sq. 

cm. 

1 

Sq. 

dm. 

(dm. 2 ) 

100 

Sq. 

(1 in . = 

1 

Sq. 

m. 

( m ■ 2 ) 

100 

Sq. 

m. = 

1 

Sq. 

Dm. 

(Dm. 2 ) 

100 

Sq. 

Dm. = 

1 

Sq. 

Hm. 

(Hm. 2 ) 

100 

Sq. 

Hm. = 

1 

Sq. 

Km. 

(Km. 2 ). 









































MEASURES OF VOLUME. 


189 


279 . Land Measure. 

Areas of small surfaces are expressed in square meters. 
Areas of land are expressed in ars. (Ar means area.) 

An ar is one square Dekameter. A cental* is, therefore, 
a square meter. 

V. Table. 

100 cental’s =1 ar (a.) 

100 ars =1 Hektar (Ha.). 

Used for same area: 

1 sq. m. =1.2 sq. yd. sq. m. and sq. yd. l S q. yd.=.84 sq. m. 

1 Hektar=2.5 acres. Hektar and acre. l A=.4 Hektar. 


280 . Measures of Volume. 


VI. Cubic Measure. 

1000 Cu. mm.= l Cu. cm. (cm. 1 2 3 4 5 6 ) 

1000 Cu. cm. =1 Cu. dm. (dm. 3 ) 

1000 Cu. dm. =1 Cu. m. (m. 3 )=l ster. 


VII. Wood Measure. 


10 decisters =1 ster (s.) 

10 sters =1 Dekaster (Ds.). 

Used for same volume: 

1 cu. m.=1.3 cu. yd. cu. m. and cu. yd. 1 cu. yd.=.76 cu. m. 

1 ster=.28 cord. ster and cord. 1 cord=3.6 sters. 


1 . What is a liter ? 

How many liters will a vessel 

2. 10 dm.X10 dm.X 10 dm.? 

3. 5 dm.XB dm.X2 dm.? 

4. 2 dm.XB dm.X4 dm.? 

5. 5 dm.XO dm.X7 dm.? 

6. B dm.X5 dm.X8 dm.? 


hold whose dimensions are 

7. 2cm.X4dm.X5dm.? 

8. 8 dm.X6 m.X7 m.? 

9. 4 m.X7 cm.Xl5 dm.? 

10. 9cm.x8cm.X2 dm.? 

11. 5 cm.XO cm.XB dm.? 



140 


THE METRIC SYSTEM. 


281. 

Metric and United 

States Measures. 


Comparison. 

1 meter 

=89.87 in. 

1 yard =.9144 meter. 

1 Kilogram =2.2 lb. Avoir. 

1 lb. Avoir. =.4536 Kg. 

1 centimeter=.3937 in. 

1 inch =2.54 cm. 

1 Kilometer =.6214 mile. 

1 mile =1.609 Km. 

1 gram 

=15.48 grains. 

1 grain =.0648 g. 

1 liter 

=1.06 qt. liq. 

1 qt. liq. =.946 liter. 

1 liter 

= .9 qt. dry. 

1 qt. dry =1.1 liter. 

1 Hektoliter =2.8875 bu. 

1 bushel =.3524 HI. 

1 Hektar =2.471 acres. 

1 acre =.4047 Hektar 

1 ster 

=.276 cord. 

1 cord =3.62 ster. 

282. 

REDUCTION. 

a. Reduce 8 Kilograms 

b. Reduce 5 yards 

to pounds. 

to meters. 

Solution.—1 Kg.=2.2 lb. 

Solution.—1 yd. = .9144 m. 


8 Kg.=17.6 lb. Ans. 

5 yd. =4.572 m. A ns 

Reduce: 


1. 

8 Kg. to lb. 

16. 91 gr. to g. 

2. 

4 m. to in. 

17. 25 qt. dry to 1. 

3. 

7 1. to qt. liq. 

18. 75 bu. to HI. 

4. 

5 Ha. to A. 

19. 38 A. to Ha. 

5. 

67 Km. to mi. 

20. 16 cords to s. 

6. 

45 g. to gr. 

21. 19 Km. to mi. 

7. 

15 s. to cords. 

22. 43 HI. to bu. 

8. 

85 m. to in. 

23. 25 Kg. to lb. Avoir. 

9. 

25 HI. to bu. 

24. 50 cm. to in. 

10. 

17 Kg. to lb. 

25. 45 mi. to Km. 

11. 

7 yd. to m. 

26. 35 s. to cords. 

12. 

6 lb. Avoir, to Kg. 

27. 25 qt. dry to 1. 

13. 

9 qt. liq. to 1. 

28. 50 Ha. to A. 

14. 

8 in. to cm. 

29. 48 lb. Avoir, to Kg. 

15. 

48 mi. to Km. 

30. 25 bu. to HI. 



METRIC AND U. S. MEASURES. 


141 


283 . Two Reductions. 


a. Reduce 7 meters to yards. 
Solution. 

1 meter = 39.37 in. 

7 meters= 275.59 in. 

= 7.65—f- yd. Ans. 


b. Reduce 5 miles to meters. 
Solution. 

1 mi. = 1.609 Km. 

5 mi. = 8.045 Km. 

= 8045 m. Ans. 


Reduce 

1. 7 m. to yd. 6. 5 mi. to m. ll. 97 Kg. to T., U. S. 

2. 89 1. to bu. 7. 8 yd. to dm. 12. 5 bu. to 1. 

3. 2 Km. to yd. 8. 9 liq. qt. to Dl. 13. 84 cm. to ft. 

4. 5 Kg. to oz. 9. 7 lb. to Dg. 14. 5 mi. to Hm. 

5. 95 g. to lb. 10. 2 A. to ars. 15. 2 HI. to pk. 


284 . If no equivalent is given, derive it from one that is 
given. 

a. Reduce 25 gallons to b. Reduce 8 lb. Troy to Kilo- 
liters. grams. 


Solution. Solution. 

1 qt. liq. = .946 1. 1 gr. = .065 g. 

4 qt. liq. = 1 gal. = 3.784 1. 5760 gr. = 1 lb. Troy = 374.4 g. 

25 gal. = 94.6 1. Ans. = .3744 Kg. Ans. 


Reduce 

1 . 25 gal. to 1. 

2. 50 ft. to m. 

3. 15 pk. to 1. 

4. 250 rd. to Km. 

5. 5 oz. Avoir, to g. 

6. 200 sq. yd. to Ha. 

7. 25 pk. to HI. 

8 . 85 ft. to m. 

9. 875 g. to lb. Avoir. 

10. 9 s. to cu. ft. 

11.8 lb. Troy to Kg. 
12. 1 sq. mi. to Ha. 


13. 1 cu. ft. to cu. dm. 

14 . 7 bu. to 1. 

15. 14 lb. Troy to Kg. 

16. 8 oz. Troy to g. 

17 . 9 ft. to dm. 

18. 95 rd. to Km. 

19. 75 sq. yd. to Ha. 

20. 55 pk. to HI. 

21. 2 Kg. to lb. and oz. Avoir. 

22. 5 Km. to mi., etc. 

23. 17 HI. to bu., etc. 

24. 90 Ha. to A., etc. 


142 


SPECIFIC GRA VITY. 


25. 7 m. to yd., etc. 28. 25 HI. to bu., etc. 

26. 7 Kg. to lb. Avoir., etc. 29. 50 Ha. to A., etc. 

27. 13 Km. to mi., etc. 30. 8 m. to yd., etc. 

285 . Volume and Weight. 

The connection between volume and weight is one of the 
chief advantages of the metric system. 

1 cubic centimeter of water weighs 1 gram. 

1000 cu. cm. (1 liter) of water weighs 1 Kg. 

1 cubic meter of water weighs 1 Ton (metric). 


Specific Gravity (sp. g.). 

Iron is 7 times as heavy as water. The sp.g. of iron is 7. 

Silver is 10 times as heavy as water. The sp. g. of silver is 10. 

Gold is 19 times as heavy as water. The sp. g. of gold is 19. 

The specific gravity of a substance is its weight when 
the unit of weight is the weight of an equal volume of water. 

If 1 cu. dm. of iron is on one scale pan, 7 cu. dm. of water (7 liters) 
will balance it. 

If 1 cu. cm. of silver is on one scale pan, 10 cu. cm. of water will 
balance it. 

If 1 cu. m. of gold is on one scale pan, 19 cu. m. of water will 
balance it. 

Since 1 cu. dm. of water weighs 1 Kg., 1 cu. dm. of iron weighs 
7 Kg. 

Since 1 cu. cm. of water weighs 1 g., 1 cu. cm. of silver weighs 
10 g. 

Since 1 cu. m. of water weighs 1 ton, 1 cu. m. of gold weighs 19 
tons. 

286 . Table of Specific Gravities. 


Alcohol.8 

Glass.... 

.... 3.4 

Mercury.... 

..13.6 

Brass.8.38 

Gold .... 

....19.36 

Milk. 

... 1.03 

Coal (bit.) 1.25 

Ice. 

.92 

Oak. 

... .84 

Copper.8.79 

Iron.. 

.... 7.21 

Pine. 

... .65 

Cork.24 

Lead. 

....11.35 

Silver.1. 

...10.47 

















VOLUME AND WEIGHT. 


148 


1 cu. cm. of alcohol weighs .8 g.; 1 liter weighs .8 Kg.; 
1 cu. m. weighs .8 T. 

1 cu. cm. of brass weighs 8.88 g.; 1 cu. dm. weighs 8.88 
Kg.; 1 cu. m. weighs 8.38 T. 

What is the weight of 1 cu. cm. of each substance in the 
table ? Of 1 cu. dm.? Of 1 cu. m.? 

For equal volumes, which weighs the most, gold or silver? 
Brass or copper ? Iron or lead ? Milk or water ? 


287 . Weights. 

1 cm. 3 =.061 cu. in. 1 cu. in.=16.4 cm. 3 , 

a. What is the weight of 1 gallon of water ? 


Solution. 

1 gal.= 231 cu. in. 1 cm. 3 of water weighs 1 g. 

1 cu. in.= 16.4 cm. 3 . 3788.4 cm. 3 of water weighs 3788.4 g. 

231 cu. in.=3788.4 cm. 3 . =3.7884 Kg. 

1 Kg. =2.2 lb. 

3.788 Kg.=8.3-p lb. Ans. 

A gallon of alcohol weighs .8, of mercury 13.6, of milk 1.03, 
as much as a gallon of water. 

What is the weight of (Answer in Kg. and lb., 2 decimals.) 


1 . A pint (28£ cu. in.) of water ? 6. 

2. A pint of milk ? 7. 

3. A quart of alcohol ? 8. 

4. A quart of mercury ? 9. 

5. A cubic foot of water ? 10. 


A cubic foot of gold ? 
A cubic foot of silver? 
A cubic foot of iron ? 
A cubic foot of ice ? 
A cubic foot of oak ? 


11 . 

12 . 

13. 

14. 

15. 

16. 


A bar of copper 50 cm.X4 cm.X2 cm.? 

A plate of glass 5 Dm.X4 Dm.X2 mm.? 

A block of coal 8 dm.X4 dm.Xb dm.? 

A cubic meter of cork ? 

A bar of lead 1 m.XB cm.X2 cm.?. 

A cubic meter of dirt whose specific gravity is 1.5 ? 



144 


THE METRIC SYSTEM. 


17. A ster of oak ? 18. A cubic foot of pine ? 

19. A brick 8 in. X4 in. X2 in. whose specific gravity is 2.9? 

20. A building stone 8 ft. X 2 ft. X 1 ft. whose specific grav¬ 
ity is 27 ? 


288 . Volumes. 

1 gram of water = 1 cm. 3 of water. 

1 Kg. of water — 1 liter of water. 

1 T. of water = 1 m. 3 of water. 

What is the volume of 28.84 grams of iron ? 

Analysis. —Since iron is 7.21 times as heavy as water. 

7.21 grams of iron=l cm. 3 of iron. 

1 gram of iron cm. 3 of iron. 

28.84 grams of iron =4 cm.3 of iron. 

What is the volume of (Answer in cm.3, 2 decimals.) 

884 g. of lead ? 

550 g. of coal ? 

500 g. of oak ? 

91.7 dg. of iron ? 

84.5 dg. of silver ? 

67.9 dg. of gold ? 

89.4 dg. of mercury ? 

8 Kg. of milk ? 

25 Kg. of mercury ? 

19. A vessel full of water weighs 5.6 Kg. The weight of 
the empty vessel is 225 g. How many liters will it hold ? 


1 . 

288.4 g. of iron ? 

8. 

2. 

872 g. of ice ? 

9. 

3. 

562 g. of silver ? 

10. 

4. 

684 g. of alcohol ? 

11. 

5, 

988 g. of gold ? 

12. 

6. 

825 g. of cork ? 

13. 

7. 

721 g. of milk ? 

14. 

How many liters in 


15. 

25 Kg. of water ? 

17. 

16. 

5 Kg. of alcohol ? 

18. 


289. REVIEW. 

1. What is a meter ? How long is it in inches ? 
Reduce 

2. 2204621 mg. to Kg. 3 . 89870482 Mm. to cm. 

4. 18 Kg. to lb. 5 . 9 yd. to m. 6. 85 gal to 1. 





THE METRIC SYSTEM. 


145 


7 . How manyars in a square Km.? 

8 . How many liters in a vessel 1 m. X 3 dm. x7 cm.? 

9 . What is the weight in Kg. and lb. of a block of coal 
2'x3'x4'? 

10. What is the volume of 100 Kg. of mercury ? 

11. What is the speed in Km. of a train going 50 miles 
per hour ? 

12. What is the weight in Kg. of 1000 gold dollars ? 

13 . A cubic foot of water weighs 1000 ounces. What is its 
weight in Kg.? 

14 . A cubic foot of gold weighs 1204 pounds. What is the 
weight of 1 cu. cm.? 

15 . What is the weight in Kg. of a bushel of wheat ? 

16. How many Km. is it from the equator to the north 
pole ? How many miles ? 

17. The distance around the earth at the equator is 24900 
miles. How many Km. in a degree of longitude ? 

18. The tropics of Capricorn and Cancer are each 23° 30' 
from the equator. How far apart are they in Km.? 

19. What advantages does the metric system of measures 
possess over the United States system ? 

20. If the speed of a train is 1 mile in 1 minute, what is 
its speed in Km. per hour ? 

21. i of the distance from New York to Chicago is 540 
miles. How far is the whole distance in Km.? 

22. A gold dollar weighs 25.8 grains. What is its weight 
in grams ? 

23. A silver dollar weighs 412.5 grains. What is its vol¬ 
ume in cu. cm.? 

24. A cubic foot of water weighs 62£ pounds. What is its 
weight in Kg.? 

25. A dry quart=67£ cubic inches. A liquid quart=57f 
cubic inches. What is the difference in cu. cm.? 


146 


THE 20th CENTURY ARITHMETIC. 


290. RATIO. 

We may compare the numbers 12 and 4 by subtraction or 
by division. Their difference is 8 ; their quotient is 8. The 
difference of two numbers is called their arithmetical ratio; 
the quotient, their geometrical ratio. As geometrical ratio 
is the more common, it is usually called simply 11 ratio.” 

291. When two like numbers are compared , their quotient is 
called, a ratio. 

It is specially to be noted that a ratio is only between 
like numbers. There is no ratio between 12 pounds and 4 
yards, or between 12 quarts and 4 books, or even between 12 
feet high and 4 feet long. 

The sign which indicates a ratio between two numbers is 
the division sign without the line ; it is placed between the 
numbers. The ratio of 12 to 4 is written 12 : 4. This ratio 
may also be written 12-^4 or J 3 2 . 

a. Since a ratio is a fraction, it may be reduced to an 
equivalent ratio having higher or lower terms. 

V-=¥-, or • 

292. 12 : 4. 

The first number, 12, is called the antecedent. 

The second number, 4, is called the consequent. 

We now have three ways of naming the terms of 12-^4 : 

As a division , 12 is the dividend, 4 the divisor. 

As a fraction, 12 is the numerator, 4 the denominator. 

As a ratio , 12 is the antecedent, 4 the consequent. 

Examples. 

1 . Compare 15 and 5 in two ways. 

2. Name the terms of 15 divided by 5 in three ways. 

3. Reduce the ratio of 18 to 12 to its lowest terms. 
Divide both terms by 6. 


REDUCTION. 


147 


Reduce the following ratios to their lowest terms : 

4. 12 : 16 7 . 88 : 22 io. 144 : 216 

5. 25 : 85 8. 40 : 56 n. 405 : 248 

6. 21 : 28 9. 86 : 24 12 . 825 : 890 

13. Reduce 8 : 5 to an equivalent ratio whose antecedent 
is 12. Multiply both terms by 4. 


whose consequent 

is 

15 

whose consequent 

is 

24 

whose consequent 

is 

45 

whose consequent 

is 

63 

whose consequent 

is 

72 

whose consequent 

is 

18 

whose consequent 

is 

35 


Reduce the following to equivalent ratios : 

14. 8 : 5, whose antecedent is 6 ; 

15. 2 : 8, whose antecedent is 10 

16. 4 : 5, whose antecedent is 12 

17. 4:7, whose antecedent is 20 

18. 2 : 9, whose antecedent is 14 

19. 5 : 8, whose antecedent is 15 

20. 8 : 5, whose antecedent is 56 

21. Express the ratio of 1 foot to 7 inches. 

1 foot=12 inches. Ratio=12 : 7. A ns. 

Express the following ratios in their lowest terms: 

22. 1 ft to 9 in. 26. $1 to 75/. 

23. 1 bu. to 20 qt. 27. 1 lb. Troy to 1 lb. Av. 

24. 1 gal. to 154 cu. in. 28. 1 yr. to 1 mo. 

25. 6 qt. to 2 pk. 29. 1 long ton to 1 short ton. 

293 . In denominate numbers, it was said that a given 

quantity of anything is determined by comparing [finding 
the ratio of] it to a definite amount of the same quantity, 
called the unit. The subject of ratio extends this method 
of comparison and makes any amount of the quantity the 
unit. 

If 3 feet is the unit, 12 feet is 4 units. 

If the weight of a cubic foot of water is the unit, my weight is 
nearly 3 units. 

If the distance light goes in a second is the unit, the distance of 
the sun is 500 units. 

If the value of an ounce of gold is the unit, a fine bicycle costs 5 
units. 





148 


THE 20th CENTURY ARITHMETIC. 


The mind does not comprehend large numbers, and a 
comparison, using a large unit, overcomes to some extent 
the difficulty of comprehending large quantities. 

294. The result of a ratio is always an abstract number. 

12z-=- 42;=8, whatever x is. 

The ratio of 12 yards to 4 yards, 12 bushels to 4 bushels, 
12 pounds to 4 pounds, or 12 of any units to 4 of the same 
units is the same as the ratio of 12 to 4. What is the ratio 
of 12 to 4 ? 


295. The different 

kinds of ratios 

are named 

resulting fractions. 



Ratio. 


Fraction. 

a. 8 : 5 

Simple. 

* 

b. 5 :8 

Inverse (of a ). 

i 

C 7 : 8 \ 

Compound. 

f x| 

d. 3£ : 5 

Complex. 

5 


DEFINITIONS. 

a. Simple. —The ratio of two integers. It is also called a 
direct ratio, or merely a ratio. 

b. Inverse. —One whose terms are transposed from the 
order they would have in the direct ratio. The inverse ratio of 
3 to 5 is written 5:3. It is also called a reciprocal ratio. 

C. Compound. —The product of two ratios. The product 
is indicated by writing the antecedents and consequents 

under each other. The value of the ratio in c is 

n i . 5x8. 

d. Complex. —A ratio one of whose terms contains a 
fraction. 

e. The square of 8 is 9; the square of 5 is 25. The ratio 
of the squares of 8 and 5 is 9 : 25. The inverse ratio of the 
squares of 8 and 5 is 25 : 9. 




RATIO. 


149 


296. Examples. 

1. What is the inverse ratio of 3 to 5 ? 

Find the value to ^ decimals of the inverse ratio of 

2. 118 : 855. 4. 57f : 67+ 6. 8 : 3.28. 

3. 231 : 2150.42. 5. 36 : 39.37. 7. 1094 : 1760. 

8. Write the compound ratio formed by the two simple 

ratios 3 to 5 and 7 to 8. 

Reduce the following to simple ratios in their lowest terms : 


5 : 

: 12 

,, 18; 

: 15 

13. 

56 : 

: 65 

9 - 8: 

: 15. 

“■ 5: 

: 26. 

26 : 

: 72. 

21 : 

: 16 

4 : 

: 7 

14. 

52 : 

: 34 

10 - 8: 

: 7. 

12 - 8: 

: 8. 

51 : 

: 70. 


15 . What kind of a ratio is 3^ : 5 ? 

16. I have two square pieces of paper. An edge of one 
piece is 3 inches; of the other, 5 inches. What is the ratio 
of their areas ? 

17 . What is the inverse ratio of the squares of 4 and 7 ? 

18. What is the reciprocal ratio of 7 : 11 ? 

19. Compare the ratio 5 : 12 and 12 : 5. 

Express the following as simple ratios : 

20. Consequent, 12+ antecedent, 3+ 

21. Consequent, |X- 2 y 2 ; antecedent, fff. 

22. Antecedent, 5 yd. 2 ft.; consequent, 3 rd. 1 ft. 

297. What is the ratio of 

a. 1 gallon to 1 bushel ? b. 1 lb. Troy to 1 lb. Avoir.? 
Solution. —1 gallon =231 cu. in. Solution. —1 lb. Troy =5760grains. 

1 bushel=2150.42 cu. in. 1 lb. Avoir. =7000 grains, 

ratio = ratio=^7 g o 

= .1074+ Ans. =\H- Ans. 

What is the ratio of (Four decimals.) 

1. One gal. to 1 bu. ? 

2. One lb. Troy to one lb. Avoir.? 

3. One gal. to one cu. ft.? 


4. 3 ft. to 12 yd.? 

5. 1 bu. to 5 qt.? 

6. 355 to 118 ? 


150 


THE mh CENTURY ARITHMETIC. 


7. One A. to one sq. mi.? 9 39.37 to 36 ? 

8. One bn. to one cu. ft.? io. 1211 to 5000? 

11. The circumference of a circle is 710 ft., and its di¬ 
ameter 226 ft. What is the ratio of cir. to diam.? 

12. The circumference of a circle is 8927 ft. and its diam¬ 
eter is 1250 ft. What is the ratio of cir. to diam.? 

13. A cubic inch of gold weighs 4886.3 grains, and a cubic 
inch of water, 253.7 grains. What is the ratio of equal vol¬ 
umes of gold and water ? The answer is the sp. g. of gold. 

14. What is the ratio of 1826211 to 5000 ? (Four decimals.) 

15. What is the ratio of a liquid quart (57f cu. in.) to a 
dry quart (67£ cu. in.)? 

298. Variation. 

There are three kinds of variation—direct, inverse, and 
joint. Arithmetic does not teach how quantities vary, but 
it can solve problems when the kind of variation is known. 

299. Direct Variation. 

T Weight varies as quantity. If 1 gallon of water weighs 83^ 
pounds, 6 gallons weigh 6 times as much. 

2. Cost varies as weight. If l pound of candy costs 30 cents, 
3 pounds cost 3 times as much. 

3. The length of a shadow varies as the height of the ob¬ 
ject which makes it. If a stick 2 feet high casts a shadow 5 feet 
long, a stick 4 times as high will cast a shadow 4 times as long. 

Definition.— Two quantities are said to vary directly if 
they are so related that when one quantity is increased the 
other is increased in the same ratio. 

a. What is the ratio of the weight of 5 gal. of water to 
the weight of 7 gal. of water ? 

Analysis.— The ratio of the quantities is 5: 7. Since the weight of 
water varies as its quantity, the ratio of the weights will be the 
same, 5:7. 




DIRECT VARIATION. 


151 


b. What is the ratio of the quantity of work a man can 
do in 3 days to the quantity he can do in 5 days ? 

Analysis.— The ratio of the times is 3:5. The ratio of the quanti¬ 
ties of work is the same, because quantity of work varies as the time 
to do it. 

Examples. 

1. What is the ratio of the weights of 3 gallons of milk 
and 5 gallons of milk ? 

2. What is the ratio of the weights of 500 bricks and 
700 bricks ? 

3. What is the ratio of the quantity of work 9 men can 
do to the quantity 17 men can do ? 

4. What is the ratio of distances a bicyclist could go at 
the rates of 7 and 10 miles an hour respectively ? 

5. What is the ratio of the distance a bicyclist could go 
in 3 hours to the distance he could go in 9 hours ? 

6. What is the ratio of the number of silver dollars in a 
bag holding 550 pounds to the number in a bag holding 1375 
pounds ? 

7. What is the ratio of the quantity of corn eaten by 15 
horses to the quantity eaten by 35 horses ? 

8. The ratio of the circumferences of circles is the same 
as the ratio of their diameters. The diameter of one circle 
is 10 inches and of another 15 inches. What is the ratio of 
their circumferences ? 

9. The height of the Washington Monument is 555 feet, 
and of the United States Capitol 300 feet. What is the ratio 
of the lengths of their shadows cast by the sun ? 

10. The weights of gold and silver vary as their specific 
gravities. The specific gravity of gold is 19.26 and of silver 
10.51. What is th6 ratio of the weight of 1 cubic foot of 
gold to the weight of 1 cubic foot of silver ? 


152 


THE mh CENTURY ARITHMETIC. 


11. The pressure of water varies as its depth. What is the 
ratio of the pressure of water at 9 feet to its pressure at 
21 feet? 

12. The areas of circles vary as the squares of their diam¬ 
eters. The diameter of one circle is 3 inches and of another 
4 inches. What is the ratio of their areas ? 

Analysis. —The ratio of their diameters is %, but the ratio of their 
areas is the square of this: f Xf=9 : 16. 

13. The diameter of one circle is 5 and of another 6 inches.' 
What is the ratio of their areas ? 

14. The distance a stone falls varies as the square of the 
time it has been falling. What is the ratio of the distance 
a stone falls in 4 seconds to the distance it falls in 7 seconds ? 

15. What is the ratio of the distance a stone falls in 6 
seconds to the distance it falls in 8 seconds ? 


300. Inverse Variation. 

If candy is 30 cents a pound, I can buy 8'pounds; if it is 
60^ per pound, I can buy only 4 pounds. 

The quantity I can buy varies inversely as the price per pound. 

If it takes 10 hours to go to New York when the train is 
traveling 30 miles an hour, it will take only 5 hours if the 
speed is 60 miles per hour. 

The time to travel a given distance varies inversely as the speed. 

Definition.— Two quantities are said to vary inversely if 
they are so related that when one is increased the other is 
diminished in the same ratio. 

a. What is the ratio of the number of bushels of wheat I 
can buy at the prices 50/ and 80/ per bushel ? 

Analysis.— The ratio of prices is 50 to 80, or %. Since the number 
of bushels varies inversely as the price, the ratio of the number of 
bushels that could be bought at 500 to the number that could be 
bought at 80.0 is the inverse ratio of 50 to 80, which is 80: 50. 



RATIO . 


158 


Examples. 

1. Two trains leave New York at the same time. The 
first travels 40, and the second 60 miles an hour. What is 
the ratio of the distances traveled in the same time ? 

2. A bin is full of corn. What is the ratio of times that 
it will last 15 and 85 horses respectively ? 

3. For the Same area the length of a rectangle varies in¬ 
versely as the breadth. What is the ratio of the breadths 
of two equal rectangles whose lengths are respectively 42 
feet and 54 feet ? 

4. The price of corn in November is 45/ and in June 65/ 
per bushel. What is the ratio of the number of bushels 
that can be bought for $100 in November to the number 
that can be bought for $100 in June ? 

5. The quantity of light that falls on your book varies 
inversely as the square of your distance from the lamp or 
gas. What is the ratio of the amounts of light when the 
distances are 4 feet and 5 feet respectively ? 

6. The quantity of heat that falls on your body varies 
inversely as the square of your distance from the stove or 
grate. What is the ratio of the amounts of heat when the 
distances are 8 and 7 yards respectively ? 

7 . The time to go from Atlanta to Chicago varies in¬ 
versely as the speed of the train. What is the ratio of the 
time when the speed is 50 miles per hour to the time when 
the speed is 40 miles per hour ? 

8. The force of gravitation varies inversely as the square 
of the distance. What is the ratio of the force at 6 miles 
to the force at 8 miles ? 


154 


THE 20th CENTURY ARITHMETIC. 


301. PROPORTION. 


Simple Proportion. 

Proportion is also called Rule of Three. 

A proportion is an equality of ratios. 

The sign of proportion is either = or : : . 
a . b. 8 : 5=6 : 10. c. 8 : 5 : : 6 : 10. 

a. is read, “three fifths equals six tenths.” 

b. is read, “the ratio of 8 to 5 equals the ratio of 6 to 10.” 
C. is read, “8 is to 5 as 6 is to 10.” This (c) is the usual way 

of writing and reading a proportion. 

The first and last terms of a proportion are the extremes 
and the other two are the means. 

a : b :: x: y. a and y are the extremes; b and x are the means. 

A great variety of problems can be solved by proportion 
and every solution depends upon the important fact that 

302. The product'of the extremes equals the product of the 
means. 

Proof. 


8 : 5 : : 6 : 10 

Z ~ TO 

Multiply both sides by 10. 

10x8 _ g 

5 

Multiply both sides by 5. 

10X8=6X5. 


a : h : : x : y. 

a x 

b y 

Multiply both sides by y. 



Multiply both sides by b. 
ay=bx. 


303. Find the value of x in the proportion 6 : 8 ::9 : x. 
Solution.— 6 : 8 :: 9 : x 

0 X _72 The product of the extremes 

X =12 equals the product of the means. 




PROPORTION. 


155 


Find the value of x to 2 decimals in the following proportions: 


1. 

8 

10 

: 12 : x. 

16. 

24 : 

16: 

: 36 : x. 

2. 

6 

8:: 

15 : x. 

17. 

18 : 

89: 

:17 

3. 

4 

10 

: 14 : x. 

18. 

19 : 

57: 

: 4 : x. 

4. 

9 

15: 

: 21 ; x. 

19. 

88 : 

11 : 

: 15 : x. 

5. 

8 

14 

:28:x. 

20. 

16 : 

10: 

: 24 : a. 

6. 

22 

: 7: 

: 18 : x. 

21. 

855 

: 118:: 22 : x. 

7. 

18 

: 9: 

: 12 : x. 

22. 

25 : 

86: 

:7 : x. 

8. 

21 

: 16 

►::51 : x. 

23. 

9 : 25:: 

12: a. 

9. 

14 

: 18 

:: 68 : x. 

24. 

421 

: 67 

:: 100 : x. 

10. 

17 

: 5: 

: 19 : a;. 

25. 

89 : 

45: 

: 10: x. 

11. 

6 : 

: 1:: 

22 : x. 

26. 

68 : 

18: 

: 1 : x. 

12. 

1 : 

: 7:: 

9 : x. 

27. 

22 : 

7:: 

355 : x. 

13. 

8 : 

16: 

:7:x. 

28. 

181 

: 49 

:: 222 :x. 

14. 

1 : 

8.14 :: 5 : x. 

29. 

121 

: 1 : 

: 247 : x. 

15. 

2 : 

19: 

: 15 : x. 

30. 

64: 

81: 

:7:x. 


304. Give the definition of a proper fraction. 

Give the definition of an improper fraction. 

Which has the greater value ? 

Could a proper fraction equal an improper fraction ? 

The proportion 6 : 9:: 8 : x may be written 

6_8 

9 x. 

Must x in this equation be a number greater or less than 8 ? 
In the above proportion, 6 :9 and 8 : x are the two ratios. 
The antecedents are 6 and 8 ; the consequents are 9 and x. 
When two ratios are equal, it is evident that if one antece¬ 
dent is greater than its consequent, the other antecedent is 
also greater than its consequent. This is a very important 
statement. 

l. Will the value of x in the proportion 8 : 10 :: 12 : x be 
greater or less than 12 ? 



156 


THE 20th CENTURY ARITHMETIC. 


2. Will the value of x in the proportion 6: 8:: 15: x be 
greater or less than 15 ? 

Let the teacher ask a similar question for each example in the pre¬ 
ceding article, 303. The answer should be given from inspection, not 
by solving 

305. Examples. 

Arrange the numbers in the following examples under the given 
condition so that their ratio will equal the given ratio. 

a. 3 and 5. Ratio, 9 : x. Condition : x is to be greater 


than 9. Ans. 

3 : 5:: 9 : x. 




Condition : x u 

i to be greater 

Condition: x is 

to be less than 

them its antecedent. 

its antecedent. 


1. 8 and 10. 

Ratio, 12 : x. 

16. 

3 and 17. 

Ratio, 9 : x. 

2. 22 and 7. 

Ratio, 3 : x. 

17. 

11 and 61. 

Ratio, 46 : x. 

3. 9 and 11. 

Ratio, 5 : x. 

18. 

17 and 43. 

Ratio, 13 : x* 

4. 3 and 4. 

Ratio, 7 : x. 

19. 

25 and 19. 

Ratio, 8: x. 

5. 17 and 5. 

Ratio, 25 : x. 

20. 

81 and 43. 

Ratio, 7 : x. 

6. 82 and 91. 

Ratio, 31 : x. 

21. 

49 and 48. 

Ratio, 25 : x. 

7. 35 and 43. 

Ratio, 47 : x. 

22. 

25 and 16. 

Ratio, 64 : x. 

8. 62 and 17. 

Ratio, 63 : x. 

23. 

16 and 49. 

Ratio, 36 : x. 

9. 91 and 64. 

Ratio, 25 : x. 

24. 

13 and 17. 

Ratio, 5: x. 

10. 42 and 56. 

Ratio, 1 : x. 

25. 

28 and 21. 

Ratio, 8 : x. 

11. 13 and 17. 

Ratio, 18 : x. 

26. 

37 and 17. 

Ratio, 49 : x. 

12. 64 and 91. 

Ratio, 65 : x. 

27. 

41 and 99. 

Ratio, 18 : x. 

13. 53 and 83. 

Ratio, 99 : x. 

28. 

86 and 49. 

Ratio, 92 : x . 

14. 27 and 6. 

Ratio, 84 : x. 

29. 

92 and 86. 

Ratio, 1 : x. 

15. 49 and 25. 

Ratio, 13 : x. 

30. 

43 and 17. 

Ratio, 40: x. 


306. Problems. 

a. I earn $6 in 8 days. How long will it take to earn $9 ? 

Analysis. —Dollars and days are here so related that when one is 
increased the other is increased in the same ratio. The ratio of $6 
to $9 is therefore the same as the ratio of 8 days to the required num¬ 
ber of days. 


PROPORTION. 


157 


Directions. —In forming a proportion, al¬ 
ways make the unknown the 4th term and the 
like term its antecedent. Then consider : 
“ From the nature of the question, should 
the 4th term be greater or smaller than the 
3d term ? If greater, put the greater of the other two numbers as 
consequent; if smaller, put the smaller as consequent. 

b. 6 gallons of water flow from a spring in 10 seconds. 
How many gallons will flow from it in 85 seconds ? 

Analysis. —Quantity of water and time 
are here so related that when one is increased 
the other is increased in the same ratio. The 
ratio of 10 seconds to 35 seconds is, therefore, 
the same as the ratio of 6 gallons to the re¬ 
quired number of gallons. 

1. I earn $6 in 8 days. How long will it take to earn $9 ? 

2. I can run 24 yards in 8 seconds. How long will it take 
me to run 88 yards ? 

3. A railway ticket for 27 miles costs 81 cents. What will 
a ticket for 69 miles cost ? 

4. 3 gallons of milk weigh 9 pounds. How much does 5 
gallons weigh ? 

5. A train travels 360 miles at the rate of 40 miles an 
hour. How far could it have gone at the rate of 60 miles an 
hour ? 

6. When the shadow of the Washington Monument is 111 
feet long, that of the Capitol is 60 feet long. The height of 
the Monument is 555 feet. What is the height of the Capitol ? 

7. The weight of 768 silver dollars is 550 pounds. What 
is the number of silver dollars in 1375 pounds? 

8. A bicyclist rides 75 miles in 11 hours. How far could 
he ride in 5 hours ? 

9. 12 cubic inches of water weigh 7 ounces. What is the 
weight of 1 cubic foot of water ? 


Solution. 

10 : 35:: 6 : x 
10.r=6x35 
£=21 Ans . 


Solution. 

6 : 9: :8 : x. 
6£=72 
£=12 Ans. 


158 


THE 20th CENTURY ARITHMETIC. 


10. The pressure of water per square inch at a depth of 2 
feet is 14 ounces. What depth will cause a pressure of 14.7 
pounds? 

11 . Two trains leave New York at the same time. The 
first travels 40, the second 60 miles per hour. When the first 
has gone 328 miles, how far has the second gone ? 

C. Two bins of the same size are full of corn. 15 horses 
eat all in the first bin in 21 days. How long will the other 
bin last 35 horses ? 

Solution. Analysis. —Number of horses and time of 

35 : 15 : : 21 : x eating are here so related that when the first 
35.r = 15X21 is increased the second is diminished in the 
x = 9. Ans. same ratio. The inverse ratio of 15 horses to 
35 horses is, therefore, the same as the direct ratio of 21 days to the 
required number of days. 

12. Two bins of the same size are full of corn. 27 horses 
eat all in the first bin in 18 days. How long will the other 
bin last 10 horses ? 

13. If it takes 12 hours for a train to go to New York 
when the train goes 30 miles an hour, how long will it take 
when the train goes 45 miles an hour ? 

14. A field 220 feet long and 198 feet broad contains 1 
acre. How broad must a field be which is 726 feet long to 
contain the same area ? 

15. A bicyclist, traveling at the rate of 8 miles an hour, 
gets to his office in 45 minutes. At what rate must he travel 
to cover the same ground in 30 minutes ? 

16. The volume of a gas varies inversely as its pressure. 
The pressure of the atmosphere is 15 pounds per square inch. 
If a boiler has 50 cubic feet of steam (a gas) in it under a 
pressure of 111 pounds per square inch, what would be its 
volume if the steam were let off into the atmosphere ? 
Its pressure would then be 15 pounds per square inch. 


PROPORTION. 


159 


17. 

-4-a- 

Pump-handle 

The power required to work a pump-handle varies inversely 
as its distance from the bolt on which the handle works 
(the fulcrum). If the power at 4 feet from the fulcrum is 
21 pounds, what must be the power at 8 feet ? 

18. If the power at 5 feet from the fulcrum is 18 pounds, 
what must it be at 2 feet ? 

19. If the volume of a gas is 22 cubic inches when the 
pressure is 51 pounds, what would be the pressure if the 
volume were 88 cubic inches ? 

20. The maximum speed of an electric car (or of any en¬ 
gine) varies as its power. If a car of 12 horse-power can go 
20 miles an hour, how fast could- a similar car having 15 
horse-power go ? 

21. When a wheel turns on its hub, the 
distance which each point on it goes, va¬ 
ries as its distance from the center. The 
inner dot is 10 inches and the outer two 
feet from the center. The distance 
around is 62.8 inches for the inner. How 
far is it for the outer ? 

22. In a Nashville grammar-school, the ratio of the num¬ 
ber of girls to the number of boys is 5 : 4. If the number of 
girls is 815, how many boys are there ? 

23. The number of teeth on the sprocket wheel of my bi¬ 
cycle is 19; the number on the rear wheel is 8. When my 
pedals are turning the sprocket wheel 88 times a minute, 
how many rotations is the rear wheel making ? 

24. A 80-foot rail for street-car tracks weighs 1800 pounds. 
What is the total weight of the rails in a mile of track ? 



3 






160 


THE 20th CENTURY ARITHMETIC. 






O 


5 



_ 


-n-ft- 


25. A flagpole on a tower is 10 feet high. 

The narrow shadow which it casts is 3 feet 
long. If the whole shadow is 24 feet long, 
how high is the tower ? 

26. The areas of triangles having the 

same al- 
ti tude 
vary as 

their bases. Find the area of the right triangle. 

27. When the difference in longitude between two places 
is 7°, the difference in time is 28 minutes. What would be 
the difference in time when the difference in longitude is 180°? 

28. If 9 pounds of pure gold make 1989 dollars, how many 
dollars will 5 pounds make ? 

29. I have enough money to ride 285 miles when the fare 
is 8 cents a mile. If it were 5 cents a mile, how far could 
I ride ? 


30. If a clock gains 8 seconds in 5 hours, how much would 
it gain in a day ? 

31. I find that a certain distance is 128 steps. One of my 
steps is 29 inches. How many yards is the distance ? 

d. A piece of work can be done by 18 men in 80 days. 
After 19 days, 5 men quit. How long will it take the rest 
of the men to finish the work ? 


Solution. —The work left would require 18 men 30—19 days. There 
are 13 men left. The question, therefore, is: “ If 18 men can do a job 
in 11 days, how long will it take 13 men to do it.” The proportion is: 
13:18 : : 11:*. 

32. A small house can be built by 17 men in 35 days. 
After 15 days, 9 men quit. How long will it take the re¬ 
maining men to finish the house ? 









JOINT VARIATION. 


161 


33. 9 men can do a piece of work in 7 days. After work¬ 
ing 2 days, 3 men quit. In how many days will the remain¬ 
ing men finish the work ? 

34. 12 men can pick the cotton on 15 acres of land in 8 
days. If 5 men quit after 3 days, how long will it take the 
remaining men to finish the picking ? 

35. If 12 men can build a house in 25 days, how many 
men must be added that the work may be done in J of 25 
days ? 

36. Sound travels 11990 feet in 11 seconds. How many 
miles (2 decimals) will it go in 60 seconds ? 

37. The earth moves 90° in its orbit in 91.31 days. In 
how many days will it make the complete revolution of 360° 
around the sun ? 

38. The statue of Rameses II., the Pharoah of the Bible, 
standing on the banks of the Nile, is 3000 years old. Its 
base is covered with 9 feet of Nile sediment, and the total 
thickness of the Nile sediment is 40 feet. If it took 3000 
years to deposit 9 feet, how long has it taken to deposit 40 
feet ? 

307 . Joint Variation. 

a. b. 


e. d. 






































162 


THE 20th CENTURY ARITHMETIC. 


If the length of a rectangle is trebled, its area is trebled, b. 
If the breadth is doubled, its .area is doubled, C. If the 
changes are made jointly, the area is increased six times, d. 

The area of a rectangle varies as its length and as its 
breadth. But the area of a rectangle equals the product of 
its length and breadth and, hence, varies as this product. 

What is the ratio of the areas of the smallest and largest 
of the above rectangles ? 

Analysis. —The ratio of their lengths is 3:9; the ratio of their 
breadths is 2:4. The ratio of their areas is the product of these 

3 • 9 

simple ratios. This product is written 9 ; ^ 

308 . When one quantity varies jointly as two or more other 
quantities , it also varies as their product. 

The solution of problems in compound proportion depends on 
this simple statement. 

Indicate the product of the simple ratios : 


a. b. 


c. d. e. 


1. 3 : 9 and 2 : 4. 

2. 4 : 7 and 3 : 5. 

3. 6 : 8 and 9 : 14 

4. 3 : 8 and 4 : 9. 

5. 5 : 8 and 7 : 11 


6. 2 : 3, 5 : 8, and 4 : 9. 

7. 4 : 7, 3 : 5, and 6 : 11. 

8. 1 : 5, 7 : 8, and 10 : 3. 

9. 2 : 7, 9 : 4, and 4 : 5. 

10. 3 : 8, 7 : 9, and 6 : 7. 


11. What does column a indicate ? 

Am. The product of the simple ratios in it. 

12. What does column b indicate ? 

13. The distance which a train goes varies jointly as the 
speed and the time traveled. What is the ratio of the dis¬ 
tance traveled by two trains which travel, one 5 hours at 51 
miles an hour, the other 7 hours at 43 miles an hour ? 

14. Express the ratio of the areas of two rectangles; one 
7 feet long 3 feet wide, the other 9 feet long 2 feet wide. 


COMPOUND PROPORTION. 


168 


15. Express the ratio of the volumes of two rectangular 
solids; one 12 feet long 5 feet wide and 8 feet thick, the 
other 9 feet long 7 feet wide and 2 feet thick. 

16. Express the ratio of the time required to build a house 
when there are 9 workmen working 10 hours a day to the 
time required when there are 15 workmen working 8 hours 
a day. 

17. Express the ratio of the times of traveling of these 
two trains : One travels 850 miles at 80 miles an hour, the 
other 200 miles at 50 miles an hour. 

18. Expiess the ratio of the quantities of corn necessary 
to last 15 horses 7 weeks and 19 horses 14 weeks. 

19. Express the ratio of the volumes of two blocks of 
granite, one 4 feet long 2 feet wide and 10 inches thick, 
the other 3 feet long 1 foot wide and 5 inches thick. 

20. The ratio of the weight of a gold dollar to the weight 
of a silver dollar is 1 to 16. The ratio of the weights of the 
same amounts of gold and silver is 19 to 10. Express the 
ratio of the number of dollars that can be made from a 
cubic foot of gold to the number that can be made from a 
cubic foot of silver. 


309 . Compound Proportion. 

A compound proportion is a proportion in which one of 
the ratios is compound. ^ ^ :: 15 : x. 

The product of the extremes equals the product of the means. 

(1) 2X3X*=9X4X15 

(2) 9X 4X15 

V ; 2X3 

From (2) we see that the value of x equals the product of 
the means divided by the product of the left extremes. We 
can avoid the work of (1) and (2) as follows : 




164 


THE noth CENTURY ARITHMETIC. 





z=90 


When all the left extremes do not cancel, perform the 
operations in (1) and (2) with the remaining factors. 

310 . Find the value of x in the following 'proportions: 


1. 

3 : 9 
2:4*' 

15 : z. 

2. 

5 : 14 . 
7:11" 

: 15 : x. 

3. 

15 : 14 
49 : 9 

:: 21 : x. 

4. 

12 : 15 
21 : 18 

:: 17 : x. 

5. 

14 : 17 
35 : 19 

:: 16 : a. 

6. 

7 : 22 
31 : 13 

:: 1 

7. 

15 : 34 
62 : 27 

:: 14 : x. 

8. 

25 : 57 
38 : 35 

:: 21 : 

9. 

38 : 49 
63 : 19 

:: 18 : z. 

10. 

45 : 13 
26 : 21 

:: 27 : x. 

11. 

7 : 53 
13 : 22 

:: 25 : x. 


12 : 16 

12. 14: 18:: 15 : x. 

4 : 5 

7 : 18 

13. 81 : 22 :: 4 : x. 

9 : 11 

19 : 25 

14. 15: 48:: 21 : 

24 : 57 

14 : 28 

15. 21 : 16 :: 24 : *. 
18 :49 

12 : 85 

16. 21 : 58 :: 86 :.x. 
16 : 42 

18 :25 

17. 45 : 86 :: 17 : x. 
84 : 21 

42 : 85 

18. 28 : 30 :: 14 : x. 
49 : 18 

43 : 7 

19. 19 : 31 :: 22 : z. 
29 : 13 

65 : 26 

20. 42 : 34 :: 45 : x. 
88 : 26 




COMPO UND PROPOR TION. 


165 


311 . In compound proportion, one quantity always varies 
as two or more quantities and, hence, as their product. The 
variation may, of course, be direct or inverse. 

a. A boy rides a bicycle 180 miles in 8 days, riding 10 
miles per hour. How far could he ride in 7 days, riding 9 
miles per hour ? 

Analysis. —The distance he could go varies jointly as to speed and 
the number of days. In forming the proportion, we consider each 
variation just as if the other were not to be considered. 


The answer is to be distance; we, therefore, make the given dis¬ 
tance the 3d term. 

First step. :: 180 : x. 


Consider the first variation : If he goes 180 miles in 3 days, he will 
go farther in 7 days. Therefore, arrange the ratio between 3 and 7 so 
that x , as far as their ratio is concerned, will be more than 180. 

8 * 7 

Second step. ’ :: 180 : x. 


Consider the second variation : If he goes 180 miles, traveling at a 
speed of 10 miles per hour, he will not go that far when traveling at a 
speed of 9 miles per hour. Therefore, arrange the ratio between 10 
and 9 so that x, as far as their ratio is concerned, will be less than 180. 

q . n 

Third step. |Q ! g • : 180 : X. 


Fourth step. Solve the proportion. 

The only difficulty in compound proportion is in making the state¬ 
ment beginning with “If” in considering each variation. Remember, 
in making the statement, that each variation is to be considered as if 
it were the only one in the problem. 

b. If the freight is $72 on 25 slabs of marble, each 17 feet 
long 8 feet wide and 6 inches thick, hauled 144 miles, 
what would the freight be on 84 slabs, each 15 feet long 2 
feet wide and 5 inches thick, hauled 860 miles ? 







166 


THE 20th CENTURY ARITHMETIC. 


Analysis. —For reference, write down the quantities as follows: 

Freight. No. Slabs. Ft. long. Ft. wide. In. thick. Distance. 

$72 25 17 3 6 144 

x 84 15 2 5 360 


Consider each step separately. As each ratio is decided upon, 
write it in the proportion. 

1. The answer is to be dollars ; make dollars the 3d term. 

2. If it costs $72 to transport 25 slabs, it will cost more to trans¬ 
port 34 slabs. Write the ratio accordingly. 

3. If it costs $72 to transport slabs 17 feet long, it will cost less if 
the slabs are only 15 feet long. 

4. If it costs $72 to transport slabs 3 feet wide, it will cost less if 
the slabs are only 2 feet wide. 

5. If it costs $72 to transport slabs 6 inches thick, it will cost less if 
the slabs are only 5 inches thick. 

6. If it costs $72 to transport slabs 144 miles, it will cost more to 
transport them 360 miles. 


25 : 34 
17 : 15 
3:2:: 72 : a; 
6 : 5 

144 : 360 


Solution.— 2 

P : n 0 

: jl : : jlfi : z=120 Ans. 

0:0 

% m : m so 


7. Solve the proportion. 

312. i. A boy rides a bicycle 180 miles in 3 days, riding 
10 miles an hour. How far could he ride in 7 days, riding 
9 miles per hour ? 

2. If the freight is $58 on 17 slabs of marble, each slab 
being 8 feet long, 3 feet wide, and 2 inches thick, what 
would the freight be on 25 slabs, each 5 feet long, 2 feet 
wide, and 3 inches thick ? 

3 . 9 workmen, working 10 hours a day, build a house in 
25 days. How long would it have taken 15 workmen, work¬ 
ing 8 hours a day ? 

4. 15 horses in 7 weeks eat 60bushels of corn. How many 
bushels will last 19 horses 14 weeks ? 



COMPOUND PROPORTION. 


167 


5. A block of granite 4 feet long, 2 feet wide, and 10 
inches thick weighs 1200 pounds. What will a block weigh 
which is 8 feet long, 1 foot wide, and 5 inches thick ? 

6. A tub has six water-faucets over it. When two fau¬ 
cets are turned on, each faucet delivers 9 gallons of water 
per minute and the tub is filled in 4 minutes. When 5 
faucets are turned on, each faucet delivers 7 gallons per min¬ 
ute. How long does it take to fill the tub ? 

7. If gas costs $2.50 a month when two jets are burned 
8 hours every night, gas being $2 per thousand cubic feet, 
how much would it cost to burn 5 jets 2 hours every night 
when gas is $8 per thousand cubic feet. 

8. 20 negroes can pick the cotton from 4 acres in 8 days 
if each negro picks 100 pounds a day. How long will it 
take 12 negroes to pick 7 acres if each negro picks 80 pounds 
a day ? 

9 . When 8 grates in a house are used, 2 tons of coal are 
burned in 25 days. How long will 7 tons of coal last when 
2 grates are used ? 

10. If 100 boys can solve 100 problems in 10 minutes, how 
long would it take 25 boys to solve 50 problems ? 

11. It cost $1050 to carpet an assembly-hall which is 90 
feet long, 51 feet wide, with carpet which is 86 inches wide, 
costing $2 a yard. What would it cost to carpet a hall 75 
feet long, 86 feet wide, with carpet which is 27 inches wide, 
costing $8 a yard ? 

12. 8 men working 10 hours a day plow 5 acres of cotton 
in 4 days, how many acres could 7 men plow in 9 days, work¬ 
ing 8 hours a day ? 

13 . The times of rowing down and up a stream are as 8 to 
2. If a man rows 7 miles down stream in 50 minutes, how 
long will it take him to row 5 miles up stream. Note — The 
time up varies jointly as the ratios of the times and distances. 


168 


THE mh CENTURY ARITHMETIC. 


14. The speeds of two bicyclists, A and B, is as 9 to 10. 
If A goes 25 miles in 75 minutes, how long will it take B to 
go 15 miles ? Note. —The time varies jointly as the speeds and 
distances. 

15. A mile of street 60 feet wide was paved with belgian 
blocks by 50 men in 85 days, working 10 hours a day. How 
long will it take 45 men to pave 8 miles of a street 40 feet 
wide, working 9 hours a day ? 

16. A teacher charges for her instruction by the hour. If 
she earns $20 in 5 days, working 4 hours per day, how much 
would she earn in 17 days, working 5 hours per day ? 

17. Walter Howard and Joe Johnson rode from Atlanta 
to Chicago, 750 miles, in 10 days, riding 13 hours a day. 
How long would it take them to go from New York to San 
Francisco, 3000 miles, riding 10 hours a day ? 

18. The freight on 200 bales of cotton from New Orleans 
to New York, 1350 miles, is $425 when the bales are not 
compressed. What would be the freight on 500 compressed 
bales from Mobile to New York, 1240 miles, the charge vary¬ 
ing as space ? A bale of cotton is 38 inches thick; when 
compressed, 14 inches thick. 

19. The first term in a compound ratio is composed of the 
factors 12, 25 and 33; the second term, 15, 22 and 42; the 
third term, 16 and 27. Find the fourth term. 

20. The weights of granite and iron are as 3 : 8. A block 
of granite 4 feet long, 2 feet wide and 10 inches thick, weighs 
1200 pounds. What is the weight of a block of iron 2 feet 
long, 1 foot wide and 8 inches thick ? 

21. At a large fire in Cincinnati 25 streams of water were 
kept going for three hours, and from each nozzle 300 gallons 
of water issued per minute. Supposing the time to extin¬ 
guish the fire varies as the number of streams on it and the 
quantity of water thrown, how long would the fire have 


PARTITIVE PROPORTION. 


169 


lasted if there had been 40 streams, each giving 250 gallons 
per minute ? 

22. A man 26 feet from a 2000 candle-power electric light 
receives on his paper a 3 candle-power light. What Avould 
be the light on his paper if he were 5 feet from a 15 candle- 
power gaslight ? Note.—I ntensity of light varies inversely as 
the square of the distance. 

23. A dog runs 960 feet in 5 minutes. How far will a rab- 
bit run in 12 minutes if the dog takes 6 leaps while the rab¬ 
bit takes 8 and two of the dog’s leaps equal three of the 
hare’s. 

24. A rabbit is 120 feet ahead of a dog and the dog runs 
360 feet per minute. How long before the dog will catch 
the rabbit, if the dog takes 6 leaps while the rabbit takes 8, 
but 2 of the dog’s leaps equal 3 of the rabbit’s ? Note.— Find 
by proportion how far the rabbit runs per minute, then solve by 
analysis. 

25. A hare takes 10 leaps while a hound takes 7, but 3 ot 
the hound’s leaps equal 5 of the hare’s. If the hare has a 
start of 280 feet, and the dog runs 350 feet per minute, how 
long before the hound will catch the hare ? 

26. Two stenographers, Misses A and B, are writing from 
dictation. A writes 3 pages while B writes 5, but A’s pages 
are twice as wide as B’s. There are 24 lines on B’s pages, 
and 32 on A’s. A writes 90 words per minute. What is B’s 
speed ? 


313 . Partitive Proportion. 

(See Article 117.) Let us compare the heights of the 
Washington Monument, the Pyramid of Cheops, and the 
Lincoln Cathedral. The heights, expressed in different 
ways, are : 



170 

THE mh 

CENTURY ARITHMETIC. 


Monument. 

Cheops 

Cathedral. 

a. 

ps of a mile. * 

tVs °f a mile. 

■fa of a mile. 

b. 

100 rods. 

150 yards. 

300 feet. 

c. 

550 feet. 

450 feet. 

300 feet. 

d. 

11 units. 

9 units. 

6 units. 


It is readily seen that the comparative heights is best un¬ 
derstood by the method of expression used in d. A com¬ 
parison of values is best made by expressing them in the same 
unit by as small integers as possible. The unit in d is 50 feet. 
In d, 11, 9, and 6 are merely representatives of the true 
heights. 

Again, we are accustomed to expressing parts by fractions, 
but, for the purpose of comparison, we also represent them 
by integers. 

60 inches --- 

--,_ 20 _,_ 28 in. _ 1 

S unils 5 units 7 units 

The line AB is divided into 3 parts which are proportional 
to 3, 5, and 7. 

A whole is equal to the sum of its parts. 

If the parts are represented by 3, 5, and 7, the whole must be rep¬ 
resented by 3 + 5+7. 

314. Partitive Proportion is the process of dividing a 

number into parts which are proportional to given numbers. 

a. Divide $60 into three parts which are proportional to 
3, 5, and 7. 

Analysis. Since the parts are represented by 3, 5, and 7, the 
whole must be represented by 15. If 15 units represents $60, 1 unit 
is $4, and 3 units is $12, etc. 

b. Divide $150 among 5 men, 4 women, and 8 boys so 
that a man may receive twice and a woman three times as 
much as a boy. 






PARTITIVE PROPORTION. 


171 


Analysis —If 1 represents what a boy Boy Man woman, 
receives, 2 will represent what a man and 12 3 

3 what a woman receives 8 will represent 
what 8 boys receive, 10 what 5 men receive, ° 

and 12 what 4 women receive. 8—j—10—|—12 will represent what all 
receive. If 80 units represent $150, 1 unit is $5. Therefore, a boy 
receives $5, a man $10, and a woman $15. 

Problems. 

1. Divide $60 into three parts which are proportional to 

3, 5, and 7. 

2. Divide $80 into three parts which are proportional to 

4, 5, and 7. 

3. Divide 510 into four parts proportional to 4, 5, 9, 
and 12. 

4 . Divide 132 into five parts proportional to 2, 3, 7, 9, 
and 12. 

5 . Divide 1242 proportional to 4, 9, 15, 20, and 21. 

6. There are four numbers proportional to 3, 5, 6, and 
10 ; the smallest is 21. What are the others ? 

7 . The relative distances from the sun of Mercury, 
Venus, the earth, and Mars is 9 : 17 23 : 35. The distance of 
the earth is 92 millions of miles. What is the distance of 
each of the others ? 

8. In 1895 the relative populations of New York, Penn¬ 
sylvania, Illinois, and Missouri was 22, 19, 15, and 10. The 
sum of the populations was 19,800,000. What was the 
population of each ? 

9. Divide $500 between A, B, C, and D so that A may 
get as much as C and D, B as much as A and C, and D twice 

as much as C. Let l=C’s share. 

10. The heights of the Great (proposed) Watkin Tower 
(England), Eiffel Tower (France), Washington Monument 
(United States), and Great Pyramid (Egypt) are propor- 


172 


THE 20th CENTURY ARITHMETIC. 


tional to 23, 19.5, 11.1, and 9. The Pyramid is 450 feet 
high. How high is each of the others ? 

11. Divide $256 into parts proportional to $, f, f, and -J. 
Note.—T he parts are proportional to 7, 5, 3, and 1. 

12. Divide $1089 into parts proportional to §, f, f, and 
Note.—R educe the fractions to common denominators. 

13. Divide $50.60 among 7 men, 4 women, and 5 boys, so 
that a boy shall receive 3 times as much as a man, and a 
woman twice as much as a boy. 

14. The dates of the following important events are pro¬ 
portional to 3.5, 74.6, 84.5, 90.75, and 94.8: Jerusalem de¬ 
stroyed by Titus, discovery of America, first newspaper in 
America, battle of Waterloo, and publication of this book. 
Find the date of each event. 

15. Divide $137.50 among A, B, and C, so that A may get 
f as much as B, and C as much as A and B together. 


315. Comparison of Numbers. 

a. 2 is what part of 3 ? 

Analysis.— 1 is % of 3, therefore 
2 is % of 3. Ans. 

b. 3 boys are what part of 5 boys ? Ans. § . 

3 sevenths are what part of 5 sevenths ? Ans. f. 

The comparison of fractions having the same denomi¬ 
nators is the same as the comparison of their numerators. 


Examples. 


1. 2 is what part of 3 ? 

2. 3 is what part of 4 ? 

3. 2 is what part of 7 ? 

4. 5 is what part of 9 ? 

5. 8 is what part of 12 ? 


6. 9 is what part of 15 ? 

7. 12 is what part of 18 ? 

8. 16 is what part of 20 ? 

9. f is what part of f ? 

10. T % is what part of T 9 3 ? 



COMPARISON OF NUMBERS . 


178 


11 . 

12. § is 


16. 


part of 



4| 

is 1 ? 

17. t 5 t ^ A? 

23. 

isf ? 

18 . A is A ? 

24. 

8f 

is f? 

19. A i® A ? 

25. 

4f 

is 1? 

20. f is f ? 

26 . 

6f 

is S? 

21 . if is A? 

27. 

14 

is § ? 22. is f ? 

f is what part of f ? 

28. 

2f 


is 24? 


is i? 


Analysis— a. Reducing them to a common denominator, the ques¬ 
tion reads: A is what part of if ? Ans. tf* 

b. “ f is what part—,” that is, § is a part of something. 

In the question, “ 2 is what part of 3,” 2 is the numerator of the 
the answer; therefore, make the part the numerator. 

I 8 

Shortened process: - — • 


Examples. 


1. 

f 

is 

what 

part 

off? 

6. 

f is what part of f ? 

2. 

f 

is 

what 

part 

of ^ ? 

7. 

f is what part of f ? 

3. 


is 

what 

part 

off? 

8. 

| is what part of if ? 

4. 

f 

is 

what 

part 

of H? 

9. 

A is what part of A 

5. 

i 

is 

what 

part 

off? 

10. 

A is what part of if 


What part of 

11. J is |? 

12. f is f ? 

13. | is f ? 

14. f is f ? 

15. T \ is I ? 

16. f is if ? 


A i s if ? 
if is 1* ? 

19. A is 


17. 

18. 


8 ? 
2 5 ‘ 


20. S§ is 2f ? 

21. 5J is 8f ? 

_Ofi 0 3 9 


23. Si is 2f ? 

24. 2J is 1^ ? 

25. 8f is 2J ? 

26. 9$ 

27. 

28. is 8}-$ ? 


9| is 7H ? 
5 t V is 4* ? 


174 


THE 20th CENTURY ARITHMETIC. 


317. To Find What Fractional Part One Denomi¬ 
nate Number is of Another. 


Directions. —Reduce both numbers to the same denomination, then 
compare them. 

What fractional part of 5 yd. 2 ft. 4 in. is 4 yd. 2 ft. 8 in.? 
5 yd. 2 ft. 4 in. =208 in. 4 yd. 2 ft. 8 in.=176 in. The question might 
now read : What part of 208 inches is 176 inches ? Evidently, 
i^f=l^=.846-(- Ans. 


What fractional part oj 


1 . 

2 . 

3. 

4. 

5. 
11 . 
12 . 

13. 

14. 


4 bu. is 3 bu.? 

6 in. is 2 in.? 

5 lb. 2 oz. is 4 lb.? 
1 long ton is 1 T.? 


6. 8 lb. is 7 lb.? 

7. 4 yd. is 1 yd.? 

8. 3 cwt. 8 oz. is 2 cwt. 80 lb.? 

„ . 9. 3 bu. 2 pk. is 1 bu. 3 pk.? 

75 yd. is 10 rd. 5 yd.? io. 2 bu. 3 pk. 7 qt. is 45 qt.? 

8 rd. 7 yd. 2 ft. is 125 ft.? 

65 pt. is 3 pk. 7 qt. 1 pt.? 

1 gal. 3 qt. 1 pt. is 1 gal. 1 pt.? 

18 cwt. 90 lb. is 16 cwt. 10 lb.? 


15. 11 mi. 156 rd. 5 yd. is 5 mi. 74 rd. 2 ft.? 

16. 8 cwt. 17 lb. is 7 cwt. 60 lb.? 

17. f ft. is | ft.? 19 . 2 rd. 3| yd. is 8£ yd.? 

18. 2f gal. is li gal.? 20. 3 gal. 2£ qt. is 2 gal. 3| qt.? 


318. Arithmetical Ratio. 

When two like numbers are compared their difference is 
called their arithmetical ratio. 

There is no special difficulty in finding the arithmetical 
ratio of any numbers except common fractions. 

The arithmetical ratio of common fractions may be 
found in two ways. The first was given in Art. 130 ; the 
second, and simpler, is to reduce them to decimals (Art, 
182), then compare. 



ARITHMETICAL RATIO. 


175 


What is the arithmetical ratio of § and A ? 


First Method. —The L. C. M. of 8 and 25 is 200. 
f—A 6 o, A = AV Ao 7o 2 o = ?ttf- 


Second Method. 

.375 
8| 3. 

| =.375 
2 9 ?=- 36 

.015 ^4ns. 


.36 
25| 9. 

75 Proof. 

150 2S5 = ^16 

150 2,00| ,03. 


Directions. —Always carry decimals, which do not end sooner, to 
four places. If the first few figures are alike, carry them two places 
beyond the last like figure, thus: 3.14159265 and 3.14159292. 


What is the arithmetical ratio of 


1. 

4 and A ? 

11. 

| and|J? 

21. 

f and | 

? 

2. 

f and f ? 

12. 

| and If ? 

22. 

f and f 

? 

3. 

4 and if ? 

13. 

A and X V ? 

23. 

t 5 t and 

6 ? 
TTJ ‘ 

4. 

A and A ? 

14. 

t 7 t and 11? 

24. 

and 

it? 

5. 

ft and it ? 

15. 

H and f ? 

25. 

il and 

it? 

6. 

J and H ? 

16. 

V- au <l fff ? 

26. 

and 

if? 

7. 

f and $■ ? 

17. 

A and ? 

27. 

| and it ? 

8. 

A and A ? 

18. 

t and H ? 

28. 

and 

H? 

9. 

A and A ? 

19. 

| and x 7 5 ? 

29. 

it and 

it? 

10. 

A and A ? 

20. 

| and H ? 

30. 

A and 

A? 


319 . Arrange the following in their order of magnitude 
(greatest first) by reducing them to decimals . 


1. 


3 5 

TS", ^4- 

6. 

7, A, It- 

11. 

4, M, §T 

2. 

4, 

5 7 

TS, 74- 

7. 

A, A, if- 

12. 

T7, A, 4 

3. 

f, 

H, §T* 

8. 

A, it, A- 

13. 

V, W, J 

4. 

3 

5? 

if, ft- 

9. 

it, if, «• 

14. 

I, 4, il¬ 

5. 

2 

7 5 

54) T7- 

10. 

T8, A, A- 

15. 

l-5 ft. It 



176 


THE 20th CENTURY ARITHMETIC. 


320. 


PERCENTAGE. 


In practical affairs, it is usually sufficiently accurate to 
represent a part by a number of hundredths. 

25 hundredths of all children die before they are 6 years old. 

83 hundredths of all people are Christians. 

1 hundredth of school children in the United States go to college. 

31 hundredths of the population of Europe and America speak the 
English language. 

50 hundredths of all people die before they are 16 years old. 

When used in this manner, another name is used instead 
of hundredths and another way of writing them is employed. 
hundredths= per cent=%. 

25 hundredths— 25 per cent=25%. 

.25=25% • (Notice that the sign % is put after the number.) 

Read the above expressions, substituting the words per 
cent for the word hundredths. 

321. Percentage treats of calculations in which a part 
is represented as hundredths of the whole. 


322. 


.25 of 1492 == 878. 

Rate. Base. Percentage. 


25% of 1492 = 878. 


The number of parts in a hundred is called the rate per 
cent or, briefly, the rate.* 

The whole is called the base. 

The part is called the percentage. 

Name the parts in 


1. 88% of 800=264. 

2. 6% of 250=15. 

3. 10% of 5280=528 

4. 12% of 875=45. 


5. 87^% of 24=9. 

6. 16|% of 42=7. 

7. 25% of 1492=878. 

8. 16f % of 1812=802. 


*The rate per cent is used in preference to any other rate because 
its use produces the least work in making calculations. 



PERCENTAGE . 


177 


323. Percentage is that part of the base expressed by the 
rate . Note.— There is very little difficulty in solving problems in per¬ 
centage when the relation of the parts to each other is understood. 

324. The rate is the ratio of the percentage to the base. 
When expressed in hundredths, it is the rate per cent. 

325. What is my rate of income when I get $2 by lend¬ 


ing $25 ? 

Analysis. —My rate of income is the ratio of 2 to 25: 

5 V=-08=8%. Ans. 

What is my rate of income when I get 

1 . $2 on $25 ? 6. 2 lb. on 7 lb.? 11. 8 gal. on 81 gal.? 

2. $8 on $50 ? 7. 4 lb. on 85 lb.? 12. 7 gal. on 46 gal.? 

3. $8 on $75 ? 8. 7 lb. on 22 lb.? 13. 5 gal. on 82 gal.? 

4. $8 on $25 ? 9. 9 lb. on 47lb.? 14. 4 gal. on 19 gal.? 

5. $2 on $2< 

326. We 


? 10 . 6 lb. on 63 lb.? 15. 9 gal. on 61 gal.? 

now have three ways of expressing 6 hun¬ 


dredths : 


Ton* 

.06 

6 %. 


As a common fraction : 

As a decimal fraction : 

As a rate per cent: 

327. Examine the following : 
a . 6%=.06 ' b. 25%=.25. 

=T§7F = 3$J. TS® i‘ 

Reduce each of the following to a common fraction : 


1. 6%. 6. 18%. 11. 90%. 

2. 25%. 7. 85%. 12. 14%. 

3. 20%. 8. 45%. 13. 15%. 

4. 16%. 9. 80%. 14. 88%. 

5. 8%. 10. 75%. 15. 45%. 

328. Examine the following : 

a. j%=.00|=.0075 b. 

Notice that f% is an( ^ that ^ 


16. 72%. 
17. 12%. 

18. 65%. 

19. 66%. 
20. 92%. 


21. 40%. 

22. 35%. 

23. 28%. 

24. 44%. 

25. 68%. 


tJ 0 —.00f—.008-J 


is 


178 


THE mh CENTURY ARITHMETIC. 


Express each of the following as a decimal and as a common 
fraction: 


1- I$%• 

2 . $%; 

3. $%; T2%- 

4. *%;$%. 


e. H%; !%• 

7. f%; A%. 

8- T 4 5 % > 2 8 7 % • 


11. 1%; 1%. 

12 . $$%; H9&- 

13. f% ; 2 9 tr% • 

14. s 9 4 %; A%- 
$%• 


16. -52% i • 

17. °lo • 

18. $$%; rg-% • 

19. $% 5 f 9 ?% • 

20. $*%; 2 4 t%- 


9. T4 % > * 

s. §% ; f%- io. ic* 


329. Examine the following : 

a. 87$%=.87$ b. 66 J%=. 66 f 

_87$_ 75 _ 66 |_200 

— 100~200 “100“800 


Reduce each of the following to a common fraction : 


1. 


6. 

16|%. 

11 . 56$%. 

16. 8 *%. 

21. 

6 f$% 

2. 

66 f%, 

7. 

624%. 

12 . 12f%. 

17. 9*%. 

22. 

8 $$% 

3. 

874%. 

. 8. 93J%. 

13. 6 f%. 

18. 8 |%. 

23. 

82$%, 

4. 

884 %, 

9. 

88 $%. 

14. 16$%. 

19. 10 j%. 

24. 

60f% 

5. 

04 %. 

10 . 

18f%. 

15. 8 3 3 2 %. 

20 . 4 t \%. 

25. 

85$% 


330. 

f of a 

, number 

is what per 

cent of it ? 




Analysis. —The question is equivalent to, 
-—- “How many hundredths of a number is % 

8 I 8 - of it?” 

Ans. % of a number is 373^% of it. 

Directions. —Reduce the fraction to hundredths. The number of 
hundredths is, by definition, the rate per cent. 

What per cent is 


a. 

b. 

c. 

d. 

e- 

f. 

i. i ? 

4? 

4? 

A? 

T V? 

A ? 

2. 4? 

4? 

1 ? 

A ? 

44 ? 

44? 

3. I? 

4? 

4? 

1 7 ? 

T 8 r 

A? 

A ? 

4. 4? 

4? 

A 7 

if ? 

44? 

44? 

5. 4? 

4? 

A? 

14? 

A? 

4S? 



PERCENTAGE. 


179 


331. To Find the 

Examine the following: 

a. What is 25% of 1492 ? 

b. What is 334% of 1776 ? 
C. What is 16f% of 1812 ? 

b. 1776 


a. 1492 __ .334 

25 592 

7460 5328 

2984 5328' 

373.00 Ann. 592.00 

What is 


1 . 15% of 1492 ? 

2. 42% of 5280 ? 

3. 50% of 1776? 

4. 75% of 3141 ? 

5. 88% of 1812 ? 

6. 65% of 2902 ? 

7. 16% of 22046 ? 

8. 10% of 160.93 ? 

9. 90% of 11.015 ? 

10. 64% of 16093? 

11. 16J% of 94625? 

12. 374% of 3937 ? 

13. 874% of 45359? 

14. 334% of 15432 ? 

15 . 124% of 31416? 


Percentage. 

(25%=.25). 

(334% = .334). 

(16f %=.16f). 

c. 1812 
•16} 

818624—1208 
10872 
1812 

is. 802.00 Ans. 

16. 45% of 1482.2? 

17. 68% of 538.61 ? 

18. 28% of 41412? 

19. 31% of 10256? 

20. 42% of 21170? 

21. 25% of 96980? 

22. 50% of 182.12? 

23. 10% of 852.48? 

24. 36% of 165792 ? 

25. 374% of 133225 ? 

26. 82f% of 814159? 

27. 5 t V% of 484821 ? 

28. 33}% of 393707 ? 

29. 66}% of 391393? 

30. 16}% of 57085? 


332. To Find the Rate. 

What part of the base is the percentage ? The answer to 
this question, when expressed in hundredths, is the rate per 
cent. 








180 


THE 80 th CENTURY ARITHMETIC. 


Examine the following : 

What per cent of 1492 is 878 ? 


.25 

14921 878. 
2984 
7460 
7460 


Analysis. —The question is equivalent to, 
“What part of 1492 is 373 ?” Evidently, tWsJ 
this, when expressed in hundredths, is the 
rate per cent. 

Ans. 878 is 25% of 1492. 


What per cent of 

1 . 1492 is 378 ? 

2 . 5280 is 264? 

3 . 1800 is 126 ? 

4 . 2950 is 177 ? 

5 . 1776 is 888 ? 

6 . 5280 is 880 ? 

7 . 365 is 83.95? 

8 . 985 is 413.7? 

9 . 555 is 88 . 8 ? 

10 . 2902 is 435.3? 


11 . 41412 is 258.825? 

12 . 31416 is 1570.8 ? 

13. 122046 is 45267.25 ? 

14. 3937 is 826.77 ? 

15. 1812 is 604? 

16. 2902 is 2539.25 ? 

17. 88 is 11 ? 

18. 414 is 276 ? 

19. 314 is 28.26 ? 

20 . 1492 is 164.12? 


333. To Find the Base. 


Examine the following : 

25% of some number is 373 ; what is the number ? 

Let x = the number, Analysis. 

. 25 x = 373 Statement. 25 % of the number = 373 

x = 3 2 7 5 3 1% °f the number = 14.92 

= i492 Ans. The number = 1492 Ans. 


What number is 

1 . 373 is 25% ? 

2 . 365 is 5% ? 

3 . 85 is 20% ? 

4 . 165 is 15% ? 

5 . 1492 is 40% ? 

6. 2120 is 124% ? 

7. 225 is 8i% ? 


t of which 

8 . 1492 is 50% ? 

9. 3141 is 374% ? 

10 . 2902 is 64% ? 

11 . 841 is 334% ? 

12 . Ill is 66 f% ? 

13. 45 is 824% ? 

14. 91 is 65 % ? 


15. 163 is 43% ? 

16. 284 is 21% ? 

17. 301 is 60% ? 

18. 365 is 18% ? 

19. 294 is 13% ? 

20 . 888 is 224 % ? 

21 . 660 is 64 % ? 






PERCENTAGE. 


181 


334. Problems Involving the Amount. 

The amount is the sum of the base and percentage. 
|oo 5 or 100% of anything is the whole of it. 

100% of the base, 1492, = Base. 

25% of the base, 373, = Percentage. 
125% of the base", 1865, = Amount. 


Examine the following: 

What number increased by 25% of itself amounts to 1865 ? 
Let x = the number. 

100% of x —j— 25% of x = 1865 Statement. 

1.25# = 1865 

x = 1492 Ans. 


What number increased by 

1. 25% amounts to 1865? 

2. 15% amounts to 414 ? 

3. 75% amounts to 1119? 

4. 88% amounts to 966 ? 

5. 85% amounts to 7128? 

6. 7% amounts to 1926? 

7. 50% amounts to 2664 ? 

8. 16|% amounts to 6160? 

9. 12£% amounts to 99 ? 

10. 66$% amounts to 690? 


11. 87$% amounts to 682 ? 

12. 381% amounts to 2416 ? 

13. 9% amounts to 342.26 ? 

14. 16|% amounts to 427 ? 

15. 6J% amounts to 2212 ? 

16. 18% amounts to 5280? 

17 . 22^% amounts to 314.16? 

18. 16% amounts to 14.14? 

19. |% amounts to 5 ? 

20. J% amounts to 37 ? 


335. Problems Involving the Difference. 

The difference is obtained by subtracting the percentage 
from the base. 

100% of the base, 1492, = Base. 

25% of the base, 373, = Percentage. 
_1 75% _ oFthe _ base^ 1119, = Difference. 







182 


THE goth CENTURY ARITHMETIC . 


Examine the following : 

What number diminished by 25% of itself gives 1119 ? 
Let x = the number. 

100% of x — 25% of x = 1119 Statement. 

.75 x = 1119 
x = 1492 Ans. 

What number diminished by 


1 . 

25% 

gives 

1119? 

11 . 

32j% gives 338.75? 

2 . 

50% 

gives 

182.5? 

12 . 

19% gives 518.4 ? 

3. 

15% 

gives 

4488? 

13. 

45% gives 342.1 ? 

4. 

6 % 

gives 

2773? 

14. 

37gives 2502.5 ? 

5. 

23% 

gives 

281.05? 

15. 

62^% gives 1501.5? 

6 . 

75% 

gives 

432? 

16. 

87^% gives 125? 

7. 

20 % 

gives 

25132 ? 

17. 

6 i% gives 265 ? 

8 . 

16J9 

o gives 4400 ? 

18. 

|% gives 12 ? 

9. 

15% 

gives 

2466.7 ? 

19. 

||% gives 32 ? 

10 . 

42% 

gives 

571.3? 

20 . 

£% gives 50 ? 


336. Miscellaneous Problems. 

1 . A bale of cotton, 500 lb., was worth $40 ; it will now 
bring only 87£% of its former value. Find the present price 
per pound. 

2 . There are 520 pupils in Crew Street school. 65% of the 
pupils are girls. How many boys are there ? 

3 . During a school year, a teacher gave out 1350 words for 
written spelling. How many words did Ella miss, her aver¬ 
age being 98% ? 

4. 56% of Scotch whisky is alcohol. How much alcohol 
(in gal.) in 1 barrel (32£ gal.) of Scotch whisky ? 

5 . Mr. Wm. James pays $225 taxes on $7500. What is the 
rate per cent of taxation ? 

6 . 2£ gallons of cream is obtained from 25 gallons of 
milk. What per cent of milk is cream ? 



PERCENTAGE. 


188 


7 . What per cent is made by buying coal by the long ton 
(2240 lb.) and selling it by the short ton (2000 lb.) at the 
same price per ton ? 

8. The best gold jewelry is marked 18 carat. What per 
cent of it is pure gold ? 

9 . What per cent of United States coins is pure gold, or 
pure silver ? 

10 . A Troy pound (5760 gr.) is what per cent of an Avoir, 
pound (7000 gr.) ? 

11 A gallon contains 281 cubic inches and a bushel 2150.4 
cubic inches. What per cent of a dry quart is a liquid quart ? 

12 . English money is 11 parts pure gold and 1 part alloy. 
What per cent of a £ is gold ? 

13. The number of pupils present in a Savannah school 
for one week is 884, which is 96% of the number enrolled. 
What is the enrollment ? 

14 . In one week a pupil spelled 185 words correctly and 
received 90%. How many words were given out ? 

15. The word and occurs in the Bible 46278 times. If 
this is 6 % of the total number, how many words are in the 
Bible ? 

16 What per cent of a leap year is the month of Sep¬ 
tember ? 

17. 334 is 88 J% of what number ? 

18. A long ton (2240 lb.) is 112% of a short ton. How 

many pounds in a short ton ? 

19 ~ 1214 ^% of a Troy pouud=oue Avoir, pound (7000 gr.). 

How many grains in a Troy pound ? 

20 . An English £, or sovereign, contains 123 grains of 
standard gold, 91f% of which is pure gold. A dollar con¬ 
tains 23.22 grains of pure gold. A £ is equivalent to how 
many dollars ? 


184 


THE 20th CENTURY ARITHMETIC. 


21. The number of pupils in a school is 495. The num¬ 
ber of boys is 25% more than the number of girls. How 
many boys are there ? 

22. A house and lot cost $6000. The place brings its 
owner 8% a year. For what does it rent per month ? 

23. A railroad reduced its wages 16$%. Mr. Bryan now 
receives $85 per month. What did he receive before the re¬ 
duction ? 

24. Mr. Dykes bought a house and repaired it at an 
expense of 8% of the cost. Taxes and insurance for a year 
amounted to 2% of the cost. At the end of a year, he sold 
it for $8000 and made $875 clear. What was the cost ? 

25. A bicycle agent’s commission is 25% of his sales. 
The expenses of his office are 10% of his sales. He sells 
bicycles at $50 each. How many bicycles must he sell to 
make $1200 ? 

26. In an examination, 20 questions were given in each 
subject. A pupil answered 16 in arithmetic, 15 in geog¬ 
raphy, 14 in history, and 12 in grammar. What was his 
average per cent ? 

27. Mr. Heard offered his house and lot for sale. There 
being no buyers, he reduced the price 10%. A purchaser 
offered 5% less than what was then asked, at which price 
Mr. Heard sold the property for $4275. What was the price 
first asked ? 


337. Gain and Loss. 

In a trade between two people, both should gain. Cubans 
produce more sugar and tobacco than they can use, and we, 
in the United States, produce and manufacture many things 
that the Cubans need. An exchange of products should 
benefit both of us. We exchange our products for furs in 
Canada ; for coffee in Brazil ; for wool in Australia; for 



GAIN AND LOSS. 


185 


wines in France, and for manufactured goods in England. 
We get what we need from them ; they get what they need 
from us ; both are gainers. Our Western States grow wheat 
and make flour ; the Southern States grow cotton ; the 
Northern States manufacture iron into farm tools, and cot¬ 
ton into cloth. Each section should gain by exchanging 
these products. I teach school, and the public gains better 
citizens ; the public pay me, and I gain my living. Arith¬ 
metic, however, deals only with those gains that can be ex 
pressed in numbers. 

When a pupil buys an arithmetic for $1.25 which cost the 
bookseller $1.00, the bookseller’s gain is 25/, but the pupil’s 
gain will depend on how he studies the book. 

338. Selling price — cost = gain. 

Cost — selling price = loss. 

In order to compare gains 'and losses, they are expressed 
as per cents of the cost. Calculations involving them are 
then easily made. 

a. John buys Atlanta Constitutions at 8 cents and sells 
them at 5 cents each ; Sam buys New York Worlds at 2 cents 
and sells them at 8 cents each. Which newsboy makes the 
better gain ? 

Analysis. —John gains at the rate of 2 on 3, or 66%%. 

Sam gains at the rate of 1 on 2, or 50%. 

339. Always calculate the rate of gain or loss on the 
cost , unless otherwise stated. 

It might be asked, “ Why not calculate the rate of gam 
on the selling price ?” The answer depends on the fact that 
when the actual gain is doubled, the rate of gain should be 
doubled also. To show this, examine the following : 



THE 20th 

CENTURY 

ARITHMETIC. 

Gain a 




Gain a 

per cent 


Selling 


per cent 

of selling 

Cost. 

price. 

Gain. 

of cost. 

price. 

$100 

$105 

$5 

.05 

■04if 

$100 

$110 

$10 

.10 

*09y X 

$100 

$115 

$15 

.15 


$100 

$120 

$20 

.20 

.16f 

$100 

$125 

$25 

.25 

.20 


Observe that when the gain is doubled, trebled, etc., the 
gain per cent of the cost is increased in the same ratio ; 
but the gain per cent of the selling price does not increase 
in the same ratio. Hence, gains or losses should always be 
expressed as per cents of the cost. 

If the relations of the parts to each other are understood 
in problems involving gains and losses, little difficulty will 
be experienced in solving them. 



340 

Find the rate per cent of gain or 

loss in the following : 


Cost. 

Selling 

price. 

Cost. 

Selling 

price. 

Cost. 

Selling 

price. 

l. 

$2 

$2.50 

8 . $16 

$18 

15 . $86.25 $90.75 

2. 

$5 

$6 

9 . $18 

$21 

16 . $16.08 

$13.25 

3 . 

$10 

$11 

io. $6 

$8 

17 . $76 

$81 

4 . 

$8 

$12 

ll. $21.10 

$16.75 

18 . $89 

$53 

5 . 

$20 

$21 

12. $43 

$45 

19 . $31 

$39 

6 . 

$12 

$15 

13 . $86.19 

$80.50 

20. $19 

$11 

7 . 

$25 

$28 

14 . $43.22 

$40 

21. $43 

$35 


341. 


Miscellaneous Problems. 



a. A house cost $1492, and was sold at a loss of 25% * 
What was the selling price ? 

The statement means that the loss was 25% of the cost. 

$1492=Cost. $1492=Cost. 

,25 = Loss %. 373 = Loss. 

$373=Loss. $1119=Selling price. Ans. 







GAIN AND LOSS. 


187 


b. A dealer makes $2 on a tennis racket, which is 40% 'of 
the cost. What was the cost ? 

Let x = the cost. Analysis. —40% of the cost = $2. 

AOx = $2 1% of the cost = $.50 

x = $5. Ans. The cost = $5. Ans . 

1. A bookseller sold a Bible for $4, which cost him $8. 
What was his gain per cent? If he had bought it for $4 
and sold it for $8, what would have been his loss per cent ? 

2. Milk tickets are 5/ each, or 25 for $1. What per cent 
on each ticket is saved by buying a dollar’s worth ? 

3. School takes in at 8:80 o’clock and lets out at 1:80, 
with 80 minutes recess. What per cent of the whole time 
is occupied by recess ? 

4. I pay $150 a year for $5000 life insurance. What rate 
per cent is this ? 

5. A grocer gains 20% by selling tea at 90/ per pound. 
What was the cost per pound ? 

6. A house is sold for $1500 and 25% of the cost is thereby 
gained. If it had been sold for $1400, what would have 
been the gain per cent ? 

7. A merchant has his goods marked 20% above cost. 
He allows a customer 10% discount on the marked price and 
sells him a hat for $2.70. What is his gain in cents ? 

8. I pay $4 a ton for coal. The man from whom I buy it 
makes a profit of 88J% ; the miner from whom he buys it 
makes 20% on the cost of mining it. What does it cost 
to mine a short ton ? 

9. I sold 2 horses at $93.75 each. On one I gained 25% 
and on the other lost 25%. How much did I lose on the 
two sales ? 

10. A broker bought cotton at 10% below the market 
price, and sold it at 5% above the market price. What was 
his gain % ? 


188 


THE 20th CENTURY ARITHMETIC. 


.11. An agent bought a bicycle and sold it for $75. 
His gain was three times .what his loss would have been 
had he sold it for $35. What was his gain per cent ? 

12. If the selling price is f of the cost, what per cent 
is lost ? 

13. If the cost is $ of the selling price, what per cent 
is gained ? 

14. A merchant bought 50 lamps at $4 each. 5 of them 
were broken in shipping ; but he sold the others at such 
a price that he gained 8% on his purchase. What was the 
selling price of each ? 

15. A suit of clothes, marked 25% above cost, was sold 
at a reduction of 20%. What was the gain or loss per cent ? 

16. The difference between 12-^% and 10% of a number 
is 8. What is the number ? 

17. A city lot was bought and increased in value 10% 
each year for two years. It was then sold for $5445. 
What did it cost ? 

18. Mr. Brittain deposited $7000 in a bank, which was 
20% of what he had. He afterwards deposited 12|% of 
the remainder. What per cent of his money was then in 
the bank ? 

19. What must a bookseller ask for a book which cost 
$3.60, in order that he may make a discount of 10% and 
still make a profit of 25% ? 

20. A book cost $1.20. What must it be marked in or¬ 
der that there may be a gain of 20% after deducting 10% 
from the marked price ? 

21. The diameter of a circle is 31.8% of the circumfer¬ 
ence. What is the circumference of a circle whose diameter 
is 1679.04 feet? 


INTEREST. 


189 


342. INTEREST. 

Money paid or received— 

(1) For the use of a house is called rent; 

(2) For the use of money is called interest. 

Interest, like rent, varies as the time for which it is 
charged. 

The money on which interest is paid is called the princi¬ 
pal. Interest is charged by the year at a rate per cent. 
Since a dollar is 100 cents, the interest (in cents ) on one dollar 
for one year is the rate per cent. The sum of principal and 
interest is called the amount. 


343. What is the interest on $25 for 8 years at 6% ? 


6% 

25 

Syr. 

$.18 

$.18 

200 

25 


$4.50 Ans. 
the interest on $1 for the 
of dollars by this. 


Analysis. —The interest 
on $1 for 1 yr. is 6^, 
on $1 for 3 yr. is 18 or $.18, 
on $25 for 3 yr. is 25X$-18. 

Directions.— The rate expresses the in¬ 
terest in cents on $1 for 1 yr. First. Find 
given time. Second. Multiply the number 


Find the interest on 


l. 

$25 

for 8 

yr. 

at 

6%. 

2. 

$36 

for 2 

y r - 

at 

6%. 

3. 

$45 

for 4 

y r - 

at 

6%. 

4. 

$78 

for 3 

yr. 

at 

6%. 

5. 

$66 

for 5 

yr. 

at 

6%. 

6. 

$58 

for 3 

yr. 

at 

5%. 

7. 

$84 

for 4 

yr. 

at 

7%. 

8. 

$92 

for 3 

yr. 

at 

8%. 

9. 

$18 

for 5 

yr. 

at 

9%. 

10. 

$45 

for 2 

yr. 

at 

8%. 

344. When 

the interest 


the interest 


ll. 

$168 

for 

8 yr. 

at 

8%. 

12. 

$235 

for 

3 yr. 

at 

6%. 

13. 

$480 

for 

2 yr. 

at 

5%. 

14. 

$540 

for 

5 yr. 

at 

7%. 

15. 

$684 

for 

2 yr. 

at 

3%. 

16. 

$642 

for 

4 yr. 

at 

8%. 

17. 

$845 

for 

6 yr. 

at 

5%. 

18. 

$920 

for 

5 yr. 

at 

4%. 

19. 

$250 

for 

3 yr. 

at 

8%. 

20. 

$860 

for 

2 yr. 

, at 

5%. 

$1 for 

• 12 months is 

6 cents, 


$1 for 1 month is i cent. 




190 


THE 20th CENTURY ARITHMETIC. 




What is the interest on $25 for 10 months at 6% ? 


6% 25 

10 mo. $.05 
$.05 $1.25 Ans. 


Analysis. —The interest 
on $1 for 1 yr. is 6 cents, 
on $1 for 1 mo. is % cent, 
on $1 for 10 mo. is 5 cents, or $.05, 
on $25 for 10 mo. is 25X$.05. 


What 

is the interest at 6% on 

1. 

$25 

for 

10 mo. ? 

2. 

$36 

for 

8 mo. ? 

3. 

$45 

for 

6 mo. ? 

4. 

$73 

for 

9 mo. ? 

5. 

$66 

for 

5 mo. ? 

6. 

$53 

for 

10 mo. ? 

7. 

$84 

for 

11 mo. ? 

8. 

$92 

for 

8 mo. ? 

9. 

$18 

for 

9 mo. ? 

10. 

$45 

for 

7 mo. ? 


11 . $168 for 11 mo. ? 

12. $285 for 6 mo. ? 

13. $480 for 5 mo. ? 

14. $540 for 4 mo. ? 

15. $684 for 8 mo. ? 

16. $642 for 8 mo. ? 

17. $845 for 5 mo. ? 

18. $920 for 9 mo. ? 

19. $250 for 11 mo. ? 

20. $860 for 7 mo. ? 




345. When the interest on $1 for 12 months is 6 cents, 
the interest on $1 for 1 month is % cent, or 5 mills. When 
the interest on $1 for 80 days is 5 mills, the interest on $1 
for 1 day is £ of a mill of 5 mills). 

What is the interest on $25 for 18 days at 6% ? 


6 % 

18 da. 


25 

$.008 


$.008 $.075 Ans. 


Analysis. —The interest 
on $1 for 1 year is 6 cents; 
on $1 for 1 day is £ of a mill; 
on $1 for 18 days is 3 mills, or $.003; 
on $25 for 18 days is 25 X $.008. 


What is the interest at 6% on 

1. $25 for 18 da.? 6. 

2. $86 for 24 da.? 7. 

3. $45 for 18 da.? 8. 

4. $78 for 12 da.? 9. 

5. $66 for 24 da.? io. 


$168 for 25 da.? 
$285 for 28 da.? 
$480 for 26 da.? 
$540 for 16 da.? 
$684 for 18 da.? 




INTEREST. 


191 


11. $58 for 18 da.? 

12. $84 for 21 da.? 

13. $92 for 27 da.? 

14. $18 for 15 da.? 

15. $45 for 27 da.? 


16. $642 for 19 da.? 

17. $845 for 14 da.? 

18. $920 for 18 da.? 

19. $250 for 29 da.? 

20. $860 for 20 da.? 


346. When the interest on $1 for 1 yr. is 6 cents, 
the interest on $1 for 1 mo. is a cent, 
and the interest on $1 for 1 da. is £ of a mill, 
What is the interest on $1 for 2 yr. 7 mo. 24 da. at 6% ? 
Q% 2 yr. 7 mo. 24 da. $1. 

$.12 =interest on $1 for 2 yr. 

,035=interest on $1 for 7 mo. 

.004=interest on $1 for 24 da. 

Ans. $.159= interest on $1 for 2 yr. 7 mo. 24 da. 


What is the interest at 6% on $1 for 


1. 2 yr. 7 mo. 24 da.? 

2. 2 yr. 8 mo. 24 da.? 

3. 4 yr. 6 mo. 18 da.? 

4. 8 yr. 9 mo. 12 da.? 

5. 5 yr. 5 mo. 24 da.? 

6. 8 yr. 10 mo. 18 da.? 

7. 4 yr. 11 mo. 21 da.? 


8. 8 yr. 8 mo. 27 da.? 

9. 5 yr. 9 mo. 15 da.? 

10. 2 yr. 7 mo. 27 da.? 

11. 3 yr. 9 mo. 25 da.? 

12. 3 yr. 6 mo. 28 da.? 

13. 2 yr. 5 mo. 26 da.? 

14. - 5 yr. 4 mo. 16 da.? 


347. 6% Method of Computing Interest. 

Directions.— First. Find the interest on the principal for the given 
time at 6%. Second. Reduce this interest to the given rate. 

The reductions may bejnade as follows : 

The interest at \°f 0 is T V of the interest at 6%. 

The interest at 1% is £ of the interest at 6%. 

The interest at 1 is i of the interest at 6%. 

The interest at 2% is £ of the interest at 6%. 

The interest at 3% is | of the interest at 6%. 




192 


THE goth CENTURY ARITHMETIC. 


Given the interest at 6%, to find the interest at 
5£%, subtract T V of itself. 6|%, add of itself. 

5%, subtract | of itself. 7%, add ^ of itself. 

4£%, subtract ^ of itself. 7£%, add | of itself. 

4%, subtract ^ of itself. 8 %, add ^ of itself, 

a. The interest on $500 for 2 yr. 7 mo. 24 da. at 6% is 
$79.50. What is the interest at 


1. 

1% ? 

4. 4% ? 

7. 7% ? 

io. 10% ? 

13. 6*% ? 

2. 

2% ? 

5. 5% ? 

8. 8%? 

ll. 4*% ? 

14. 7£% ? 

3. 

8%? 

6. 6% ? 

9. 9%? 

12. 5i% ? 

15. 12% ? 


b. The interest on $5280 for 2 yr. 5 mo. 18 da. at 6% is 
$781.44. What is the interest at the rates in a ? 

C. The interest on $1000 for 1 yr. 9 mo. 12 da. at 6% is 
$107. What is the interest at the rates in a ? 


348. What is the interest on $5280 for 2 yr. 5 mo. 18 da. 


at 8% ? 

2 yr. 5 mo. 18 da. 5280 
.12 $.148 

.025 42240 

.008 2112 

.148 528 

$781,440 

260.48 

$1041.92 Ans 


Analysis. —The interest on $1 for 
2 yr. 5 mo. 18 da. at 6% is $.148. 
The interest on $5280 at Q% is 5280X 
$.148. $781.44 is the interest at 6 %; 

K of $781.44 is the interest at 2% t 
The sum of $781.44 and $260.48 is 
the interest at 8%. 


Note. —It is practically easier in most cases to reduce the whole 
interest from 6% to the given rate, rather than to reduce the interest 
on $1 to the given rate before multiplying* 

What is the interest on 

1. $850 for 2 yr. 5 mo. 18 da. at 7% ? 

2. $780 for 2 yr. 8 mo. 24 da. at 5% ? 

3. $650 for 4 yr. 6 mo. 18 da. at 8% ? 

4. $840 for 8 yr. 9 mo. 12 da. at 4% ? 








EXACT INTEREST . 


193 


5 . 

$950 

for 

5 yr. 

5 mo. 

24 da. at 3% ? 

6 . 

$835 

for 

3 yr. 

10 mo 

. 18 da. at 7% ? 

7 . 

$785 

for 

4 yr. 

11 mo 

. 21 da. at 8% ? 

8 . 

$655 

for 

3 yr. 

8 mo. 

27 da. at 9% ? 

9 . 

$955 

for 

5 yr. 

9 mo. 

15 da. at 5% ? 

10 . 

$485 

for 

2 yr. 

7 mo. 

27 da. at 10% ? 

11 . 

$835 

for 

3 yr. 

11 mo 

. 25 da. at 4^% ? 

12 . 

$692 

for 

3 yr. 

6 mo. 

28 da. at 51% ? 

13 . 

$843 

for 

2 yr. 

5 mo. 

26 da. at 6|% ? 

14 . 

$652 

for 

5 yr. 

4 mo. 

16 da. at 7-|% ? 

15 . 

$843 

for 

2 yr. 

8 mo. 

16 da. at 8^% ? 

16 . 

$659 

for 

Syr. 

7 mo. 

28 da. at 4|% ? 

17 . 

$838 

for 

5 yr. 

9 mo. 

23 da. at 5^% ? 

18 . 

$647 

for 

2 yr. 

8 mo. 

11 da. at 9% ? 

19 . 

$749 

for 

6 yr. 

11 me 

». 9 da. at 7% ? 

20 . 

$989 

for 

5 yr. 

6 mo. 

5 da. at 6£% ? 

21 . 

$646 

for 

4 yr. 

10 mo 

». 4 da. at 7% ? 

22 . 

$737 

for 

3 yr. 

9 mo. 

19 da. at 5% ? 

23 . 

$647 

for 

3 yr. 

11 me 

>. 16 da. at 7% ? 

24 . 

$943 

for 

2 yr. 

7 mo. 

14 da. at 7|% ? 

25 . 

$741 

for 

5 yr. 

9 mo. 

19 da. at 8% ? 

26 . 

$684 

for 

4 yr. 

5 mo. 

21 da. at 8-|% ? 

27 . 

$931 

for 

1 yr. 

8 mo. 

29 da. at 9% ? 

28 . 

$682 

for 

3 yr. 

6 mo. 

6 da. at 9^% ? 

29 . 

$485 

for 

2 yr. 

10 me 

>. 9 da. at 5% ? 

30 . 

$793 

for 

4 yr. 

9 mo. 

15 da. at 5^% ? 


349. Exact Interest. 

The 6% method of computing interest is based upon the 
supposition that there are 360 days in a year ; the results 
thus obtained are, therefore, not strictly accurate. The 
United States Government, most bankers, and some business 
men calculate the exact interest by using the exact time. 



194 


THE 20th CENTURY ARITHMETIC. 


By the 6% method, The exact interest 

we say that the interest for 365 days is 6 cents, 

for 360 days is 6 cents, for 1 day is of 6 cents, 

or for 360 days is fff of 6 cents, or 

vf of 6 cents. ^fof 6 cents. 

The interest by the 6% method for any number of days 
is, therefore, of itself too much. ' 


350. Find the exact interest on $5280 from Feb. 22, 
1896, to July 4, 1898, at 8%. 

1898 7 4 

1896 2 22 

2 132 

— 2Mf (Art. 240). 

Directions. — First, Find the ex¬ 
act time in years. Second. Find 
the interest on the principal for 1 
year at the given rate. Third. 

Multiply this by the number of 
years. 

Find the exact interest on 

1. $850 for 2 yr. 168 da. at 7%. 

2. $780 for 2 yr. 264 da. at 5%. 

3. $650 for 4 yr. 198 da. at 8%. 

4. $840 for 3 yr. 282 da. at 4%. 

5. $950 for 5 yr. 174 da. at 3 %. 

6. $835 for 3 yr. 318 da. at 7%. 

7. $785 for 4 yr. 351 da. at 8%. 

8. $655 for 3 yr. 267 da. at 9%. 

9. $955 for 5 yr. 285 da. at 5%. 

10. $435 for 2 yr. 237 da. at 10%. 

11. $1492 from Dec. 13, 1895, to Dec. 23, 1896,* at 7%. 

12. $1776 from Oct. 11, 1895, to Dec. 9, 1896, at 8%. 


$5280 

.08 

Int. for 1 yr., 422.40 
_ 

Int. for 132 da., 152.76 
Int. for 2 yr., 844.80 
Int. for 2 yr. 132 da., $997.56 Ans. 


Note— 1896 was a leap year. 







TO FIND THE RATE. 


195 


13. $1812 from Sept. 29, 1894, to Feb. 29, 1896, at 5%. 

14. $1845 from May 18, 1895, to Feb. 29, 1896, at 4%. 

15. $1860 from July 7, 1894, to Jan. 19, 1896, at 9%. 

351. The five parts to an interest problem are Principal, 

Interest, Amount, Rate, and Time; if any three (except P., 
I., and A.) are given, the other two may be found. The rela¬ 
tions between these five parts should be thoroughly understood. 

The rate is the interest on $1 for 1 year. 

The interest for 1 year is that part of the principal repre¬ 
sented by the rate. When the rate is 6%, .06 of the principal is 
the interest for 1 year. 

The amount is the sum of principal and interest. 


352. To Find the Rate. 

a. When a certain sum of money is lent at 1%, the in¬ 
terest is $8 ; at what rate should it be lent to make the 


interest $18 ? 


When money 

is lent 

at 1 

per 


cent , 

the interest 

is 


1 . 

$3; 

6. 

$12; 

11. 

$15; 

2. 

$5; 

7. 

$7; 

12. 

$16; 

3. 

$6; 

8. 

$8; 

13. 

$20; 

4. 

$4; 

9. 

$11; 

14. 

$35; 

5. 

$9; 

10. 

$2; 

15. 

$25; 


b. When the interest on 
$24, what is the interest at 
terest be $36 ? 


When money is lent at 6 per 
cent , the interest is 

16. $24; 21. $48; 26. $90; 

17. $18; 22. $54; 27. $84; 

18. $30; 23. $66; 28. $96; 

19. $36; 24. $72; 29. $78; 

20. $42; 25. $12; 30. $60; 


at what rate should it be lent to 
make the interest 

1. $18? 6. $48? 11. $45? 

2. $35? 7. $49? 12. $80? 

3. $30? 8. $72? 13. $100? 

4. $28? 9. $33? 14. $245 ? 

5. $36? 10. $13? 15. $175 ? 
certain sum of money at 6% is 
oj 0 ; at what rate would the in- 

at what rate should it be lent to 
make the interest 

16. $36 ? 21. $32 ? 26. $60 ? 

17. $27 ? 22. $27 ? 27. $70 ? 

18. $45 ? 23. $44 ? 28. $72 ? 

19. $42 ? 24. $54 ? 29. $39 ? 

20. $28 ? 25. $11 ? 30. $95 ? 



196 


THE 20th CENTURY ARITHMETIC. 



353. What rate on $5280 will give $1041.92 interest in 
2 yr. 


5 mo. 

18 da.? 



5280 

Analysis. —$130.24 is the interest at 

.12 

.148 

1 % ; $1041.92 is the interest at 8% 
because it is 8 times as much as 

.025 

42240 

.003 

2112 

the interest at 1%. 

Directions. — First. Find 

.148 

528 

the interest on the prin- 


61781.440 

.08 Ans. cipal at 1% for the time. 


130.24 

| 1041.92 Second. Divide this 

1041 92 into S iven interest. 


Time. 

Principal. Interest. Yr. mo. da. 

1. $5280, $1041.92, 2 5 18. 

2. $850, $220.15, 8 8 12. 

3. $975, $165.75, 

4. $650, $92.95, 

5. $750, $140.25, 

6. $1240, $105.09, 

7. $8680, $801.68, 

8. $4258, $387.85, 

9. $5631, $1119.94, 2 

10. $4280, $1555.07, 3 


1 10 20 . 
2 7 6. 
2 12 . 
9 27. 
6 22 . 
3 19. 
5 25. 
7 18. 


Time. 

Principal. Interest. Yr. mo. da. 

11. $280.75, $62.89, 3 2 12. 

12. $315.40, $142.14, 5 7 18. 

13. $416.26, $182.98, 8 9 15. 

14. $620.35, $417.03, 7 5 19. 

15. $575.38, $221.39, 9 7 13. 

16. $1152, 

17. $1280, 

18. $1560, 

19. $2840, 

20. $5280, 


$403.20, 3 10 20. 
$263.04, 2 3 12. 
$183.04, 1 
$607.76, 3 
$716.32, 2 


5 18. 

6 24. 
5 18. 


354. 


To Find the Time. 

a. When a sum of money is lent for 1 year, the interest is 
$4 ; for what time should it be lent to make the interest $20 ? 
When money is lent for 1 year for what time should it he lent 


the interest is 


to make the interest 


1. 

$4; 

6. 

$12; 

n. 

$6; 

l. 

$20 ? 

6. 

$48? 

ll. 

$15? 

2. 

$5; 

7. 

$7; 

12. 

$14; 

2. 

$15 ? 

7. 

$49? 

12. 

$63? 

3. 

$6; 

8. 

$8; 

13. 

$16; 

3. 

$42 ? 

8. 

$72? 

13. 

$36? 

4. 

$9; 

9. 

$11; 

14. 

$20; 

4. 

$36 ? 

9. 

$33? 

14. 

$25? 

5. 

$7; 

10. 

$2; 

15. 

$18; 

5. 

$21 ? 

10. 

$13? 

15. 

$42? 









TO FIND THE TIME. 


197 


16. When a sum of money lent for 2 years brings $8 in¬ 
terest, for what time should it be lent to bring $28 interest ? 

17. When money lent for 5 months brings $15 interest, 
how long must it be lent to bring $27 interest ? 

18. YY hen money lent for 7 days brings 21/ interest, how 
long must it be lent to bring 89/ interest ? 

19. \\ hen money lent for 8 months brings 75/ interest, 
how long must it be lent to bring $1.05 interest ? t 

20. When money lent for 8 months brings $1.12 interest, 
how long must it be lent to bring $2.17 interest ? 

355. In what time will the interest on $5280 be $1041.92 
at 8%? 

Analysis. — $422.40 is the interest for 1 
year; $1041.92 is the interest for 2.4% 
years, because it is 2.4% times as much 
as the interest for 1 year. 

2-4% years =2 years 5 months 18 days. 

Directions.— First. Find the interest on 
the principal at the given rate for 1 year. 
Second. Divide this into the given interest. 

Find the time in the following : 


Principal. 

Interest. 

Rate. 

Principal. Interest. 

Rate 

l. $5280, 

$1041.92, 

8%. 

ll. $895, 

$165, 

5%. 

2. $1792, 

$80.64, 

6%. 

12. $1092, 

$180, 

6%. 

3. $2204, 

$289,706, 

7%. 

13. $1562, 

$150, 

7%. 

4. $4382, 

.$1388.26, 11%. 

14. $1846, 

$225, 

8%. 

5. $978.60, 

$518,658, 

5%. 

15. $2580, 

$360, 

9%. 

6. $4120, 

$1091.80, 

5%. 

16. $6842, 

$1000, 

5%. 

7. $3180, 

$366.05, 

4%. 

17. $3540, 

$500, 

6%. 

8. $2875, 

$856.27, 

6%. 

18. $1776, 

$112, 

7%. 

9. $1127, 

$53.88, 

4%. 

19. $1492, 

$250, 

8%. 

io. $1873, 

$146.88, 

6%. 

20. $1812, 

$175, 

9%. 


$5280 A ns. 

.08 2.4f 

422.40| 1041.92 
8448 
19712 
16896 
2816 




198 


THE 20th CENTURY ARITHMETIC. 


356. To Find the Principal. 

a. If I lend $1 for 6/, how many dollars will I lend 
for 18/ ? 

b. If I lend $1 for 6/, how many dollars will I lend for 

1. 80/? 2 . 42 /? 3 . 12 /? 4. 54/? 5.24/? 6.72/? 

C. If I lend $1 for 5/, how many dollars should I lend for 
7. 20/? 8.80/? 9. $1? 10.45/? 11.80/? 12.95/? 

d. If I lend $1 for 8/, how many dollars should I lend for 
13. $1.04? 14. $2? 15 . $1.52? 16 . $8.76? 17 . $3.86? is. $1.88? 

e. If I lend $4 for 24/, for how much should I lend $1 ? 
How many dollars should I lend for 

19. 42/? 20. $1.23? 21. $2.73? 22. $5.14? 23. $10? 24. $15? 

f. If I lend $7 for $1.40, how many dollars should I lend 
for 

25. $5? 26. $12? 27. $6.80? 28. $9.40? 29. $15? 30. $20 ? 

357. What principal will give $1041.92 interest in 2 yr. 
5 mo. 18 da. at 8% ? 

2 yr. 5 mo. 18 da. Let x = the principal. Analysis. —$.197)4 is 

•12 -197)4 x — 1041.92 the interest on $1 


-025 .592 a: = 3125.76 

for 2 yr. 5 mo. 18 

.003 

x = 5280 Ans. 

da. at 8%; the in¬ 

.148 

.592 | 3125.76 

terest on x dollars 

•049)4 

2960 

is $.19734 which 

-197^ 

1657 etc. 

equals $1041.92. 

Directions. — First. 

Find the interest on $1 for the given time at 

the given rate. Second , 

Divide this into the given interest. 

Find the principal 

in the following : 


Interest. 

Time. 

Rate. 

1. $1041.92 

2 yr. 5 mo. 18 da. 

8%. 

2. $105.09 

2 yr. 9 mo. 27 da. 

3%. 

3. $387.85 

1 yr. 3 mo. 19 da. 

7%. 

4. $165.75 

1 yr. 10 mo. 20 da. 

9%. 

5. $182.98 

8 yr. 9 mo. 15 da. * 

5%. 





THE AMOUNT. 


199 



Interest. 



Time. 


Rate. 

6 . 

$62.89 

3 

yr- 

2 

mo. 

12 da. 

7%. 

7. 

$607.76 

3 

yr. 

6 

mo. 

24 da. 

6%. 

8 . 

$183.04 

1 

yr. 

5 

mo. 

18 da. 

8%. 

9. 

$140.25 

2 

yr. 

2 

mo. 

12 da. 

84% - 

10 . 

$301.68 

1 

yr. 

6 

mo. 

22 da. 

54%. 

11 . 

$137.74 

2 

yr. 

4 

mo. 

12 da. 

6%. 

12 . 

$165.48 

2 

yr. 

4 

mo. 

24 da. 

7%. 

13. 

$318.78 

3 

yr. 

3 

mo. 

18 da. 

7%. 

14. 

$299.70 

2 

yr. 

8 

mo. 

12 da. 

10%. 

15. 

$247.38 

1 

yr. 

10 mo 

. 24 da. 

7%. 

16. 

$300. 

1 

yr. 

7 

mo. 

6 da. 

5%. 

17. 

$182.40 

1 

yr. 

3 

mo. 

6 da. 

4 4%. 

18. 

$330.96 

2 

yr. 

4 

mo. 

24 da. 

84%. 

19. 

$592.02 

3 

yr. 

7 

mo. 

6 da. 

64%. 

20 . 

$152.90 

1 

yr. 

5 

mo. 

18 da. 

74%- 


358. Problems Involving Amount. 

The amount is the sum of the principal and interest. 
What principal will amount to $6321.92 in 2 yr. 5 mo. 
18 da. at 8% ? 

100% of the principal = Principal 
19.7%% °f the principa l = I nterest 
119.7%% of the principal = Amount = $6321.92 
Let x = the principal 
1.197%* = 6321.92 

x = 5280 Am. 

Directions.— First . Find the interest on $1 for the given time at 
the given rate. Second. Divide 1 plus this into the given amount. 


Find the principal in the following : 

Amount. Time. Rate. 

1 . $6321.92 2 yr. 5 mo. 18 da. 8%. 

2. $1188.95 2 yr. 7 mo. 24 da. 4%. 

3. $1338.75 3 yr. 9 mo. 18 da. 5%. 

4. $1373.50 1 yr. 11 mo. 10 da. 6%. 





200 


THE 20th CENTURY ARITHMETIC. 



Amount. 


Time. 

Rate. 


$1542.50 

3 

yr. 7 mo. 6 da. 

64 %. 

6. 

$1886.50 

2 

yr. 8 mo. 12 da. 

84%. 

7. 

$1787.15 

1 

yr. 9 mo. 18 da. 

84 %. 

8. 

$2080.65 

2 

yr. 10 mo. 24 da. 

9%. 

9. 

$2058.50 

3 

yr. 3 mo. 18 da. 

84 %. 

10. 

$2075.52 

2 

yr. 0 mo. 9 da. 

4%. 

11. 

$761.15 

1 

yr. 10 mo. 24 da. 

9%. 

12. 

$842.55 

2 

yr. 3 mo. 18 da. 

10%. 

13. 

$983.25 

2 

yr. 2 mo. 20 da. 

6f%. 

14. 

$993.60 

2 

yr. 7 mo. 6 da. 

4%. 

15. 

$1123.65 

2 

yr. 8 mo. 12 da. 

5%. 

16. 

$2244. 

3 

yr. 3 mo. 18 da. 

84%. 

17. 

$2568. 

3 

yr. 4 mo. 15 da. 

10%. 

18. 

$1469.31 

3 

yr. 5 mo. 21 da. 

4%. 

19. 

$1402.50 

2 

yr. 0 mo. 12 da. 

6%. 

20. 

$1965. 

1 

yr. 0 mo. 24 da. 

44%. 


359. Promissory Notes. 

When one person borrows money from another, he gives 
to the lender a written promise to pay back the money ; 
this written promise is called a promissory note. 

360. “A law is a rule of action enforced by penalties 
commanding what is right and prohibiting what is wrong.” 
Just as all States have a law prohibiting theft, so they all 
have a law commanding that a man shall keep his legal 
promises. For business purposes and on account of the 
ease with which a promise can be proved when it is in 
writing, money is borrowed by giving a promissory note. 
These notes are so common that States have passed laws 
prescribing the data a note must contain in order to make 
the promise binding. 



PROMISSORY NOTES. 


201 


361. Promissory Note. 


■Mi 

1 $ Atlanta, Ga. Se P u ,8 ’ J895- 

.... 

iilli.fi 

iiviiiai 

-!•! 

.... 

Ninety days after date, I promise to pay 

1 to the order of George Washington, 

|| Six hundred twenty-five . --—— yy Dollars 

| at The Lowry Banking Company. 

; Value Received, with inte rest at 8%. 

liil 

! "|Sfn 1 Robinson Crusoe 



A promissory note must contain (1) the names of the 
parties to the note, (2) the words “ value received ,” (3) the 
sum of money {expressed in words) for which the note is given, 
and (4) the time which may elapse before the promise 
must be fulfilled. If the note bears interest, it must con¬ 
tain (5) the words “with interest .” If the rate is not 
specified, the legal rate in the State where the note is made 
is the rate of the note. Examine the above note for these items. 

The sum promised is called the face of the note. 

The person who makes the promise is called the maker. 

The person to whom the money is promised is called the 
payee. Name these features in the above note. 

362. Notes may be worded in three ways : 

1. Pay to George Washington. 

2. Pay to George Washington or bearer. 

3. Pay to the order of George Washington. 

Number 1 means that the maker need not pay what he 
promised to any one but George Washington. 

Number 2 means that the maker must pay whoever pre¬ 
sents the note when it becomes due. 

Number 3 means that if George Washington orders the 
maker to pay some one besides himself then the maker must 


















202 


THE 20th CENTURY ARITHMETIC. 


do so. The payee writes this order on the back of the note. 

Numbers 2 and 8 make the note negotiable, because the 
note can be transferred from one person to another. 

Number 8 is the customary and most common form. 

363. When the holder of a note sells it, he must write 
his name on the back, if it is of the third form ; by doing so 
he guarantees its payment. This is called endorsing the 
note. 

The endorsement may be worded in any of the three ways 
of making a note: (1) Pay to John Smith; (2) Pay to 
John Smith or bearer; (8) Pay to John Smith or order. 

364. A note is payable at the time specified ; but the 
laws of most States allow three days of grace, at the end of 
which time the note is said to mature; it then becomes due 
and must be paid. If the note bears interest, interest must 
also be paid for the three days of grace. 

365. If the maker refuses to pay according to his promise, 
the holder protests according to a form of law and the en¬ 
dorsers are notified that the note is due and unpaid ; the en¬ 
dorsers then become responsible for its payment. 

366. The dates on which a note is payable and matures 
are indicated thus : Dec. 17/20, 1895. 

367. Calculate the amount due at maturity on the fol¬ 
lowing notes : (Use days of grace for each.) 

1 . 


°jo^o 

$ / 250 Springfield, Ill. My 7 - J 89 s : 

Sixty days after date. 1 promisc to pay 

to the order of J°^ n Adams, 

Twelve hundred fifty — - — - oo Dollars 

100 

at The First National Bank. 

Value Received, with interest at 6%. 

]Vj n 2 David Copperfield. 


















PROMISSORY NOTES. 


203 


2 . 



$ Austin, Texas, _MiZ^J 89 6 - 

Five months after ^{£W promisc t0 pay 

to the order of Thomas Jefferson, 

Eight hundred sixty-five —-.---— Q o 

" Ml —---—-JLyOliarS 

The First National Bank. 

Value Received, with interest at 7%. 

No. 3 Daniel Deronda. 

3 . 

a 

a 

a 

a 

a 

$ 51 6 - Tackson, Miss. August 3, ^96. 

Ninety days ajter^ date, 1 prQmise to pay 

to the order of James Madison, 

Five hundred thirty-six --—- 50 

-----—-- lyUildl u 

. 100 

The First National Hank. 

Value Received, with interest at 8%. 

a 

No* 4 J°^ n Gilpin. 

4 . 

H^rP 

■^rP 

•4&P 

’^rP 

•'^rP 

*^rP 

’^rP 

•^rP 

*^rP 

*^rP 

•^rP 

*^rP 

'^rP 

‘^rP 

•^rP 

'^rP 

•^rP 

$ 87 ~ Montgomery, Ala* J uly 17 > \%9 6 - 

One hundred days after date, I prnrni ^ tn pay 

to the order of James Monroe, 

Eighty-seven --—---—- oo rwu-c 

---i^unars 

100 

The First National Hank. 

Value Received, with Merest at 7%. 

No. 5 Felix Holt. 










































204 


THE 20th CENTURY ARITHMETIC. 


5. 



$ 128 - Columbia, S. C. October ,6 ’ 189 6 - 

Sixty days after date, I pmrnisp fn pay 

tn tho nrApr nf J ohn Q W Adams - 

rS 

One hundred twenty-eight ---— 65 Dollars 


100 

af The First National Bank. 


Value Received, with interest at 7 % - 


]\j n 6 Henry Little. 


368. What is the amount at maturity of each of the follow¬ 
ing notes? (Use days of grace for each.) 


Date. 

Face. 

Rate. 

Time. 

l. Jan. 15, 1896 

$788.30 

7 % 

Ninety days. 

2. Feb. 29, 1896 

$682.90 

5 % 

Sixty days. 

3. March 16, 1896 

$1289 

4 % 

Thirty days. 

4. April 17, 1896 

$75 

8% 

100 days. 

5. May 18, 1896 

$681.20 

9% 

Ninety days. 

6. June 19, 1896 

$1841 

64% 

Sixty days. 

7. July 20, 1896 

$67.52 

74% 

Thirty days. 

8. Aug. 21, 1896 

$841 

84 % 

100 days. 

9. Sept. 22, 1896 

$2384 

54 % 

Ninety days. 

10 . Oct. 28, 1896 

$1776 

64 % 

Sixty days. 


369. • Annual Interest. 

When interest is charged for only one period of time, as 
in the notes already given, it is called simple interest. 

370. When it is stated in the note that the interest is 
payable annually, such a note is said to bear annual in¬ 
terest. 

371. Annual interest should be paid, according to the 
agreement, one year from the date of the note, two years 















ANNUAL INTEREST. 


205 


from the date of the note, etc. Should the borrower fail to 
pay the annual interest when it becomes due, the laws of 
some States permit the lender to charge simple interest on 
the unpaid annual interest. Some States permit the rate 
of this simple interest to be that expressed in the note; 
others, the legal rate. 

In the following examples use the rate in the note. 

372. 


ojjo^c 

8ocPA 

Chtragn, Ill. 0°^ J 4> 139 2 - 



On demand, I p rom ise to pay 


to the order of 

Andrew Jackson, 


Eight hundred 

OO nr.Hflrc 


Value Received, 

100 

with interest annually at 6%. 


No. 7 

Nicholas Nicklehy. 


No interest having been paid on this note, what is the 
amount of it 3 yr. 6 mo. after date ? (No grace.) 


Analysis. —The annual interest is $48. 

The 1st installment remains unpaid 2 yr. 6 mo. 

The 2d installment remains unpaid 1 yr. 6 mo. 

The 3d installment remains unpaid 6 mo. 

The simple interest on the annual interest for 4 yr. 6 mo. 
will give the interest allowed on overdue annual installments. Add 
this to the amount found at simple interest and it gives the amount 


at annual interest. 

3 yr. 6 mo. $800 
.18 .21 

.03 168.00 

12.96 
800.00 


4 yr. 6 mo. $48 

.24 .27 

.03 336 

.27 96 


$12.96 


Amount $980.96 Arts. Note. — When the interest is 
paid annually, as is customary, 
this calculation does not apply. 



















206 


THE 20th CENTURY ARITHMETIC. 


373. No interest being paid on the following notes , what is 
the amount of each at annual interest ? (No grace.) 



Face. 

Time. 

Rate. 


Face. 

Time. 

Rate. 

1 . 

$800 

3 yr. 6 mo. 

6%. 

11. 

$850 

2 yr. 5 mo. 18 da. 

7%. 

2. 

$168 

3 yr. 11 mo. 

8%. 

12. 

$780 

2 yr. 8 mo. 24 da. 

5%- 

3. 

$235 

3 yr. 6 mo. 

6 %. 

13. 

$650 

4 yr. 6 mo. 18 da. 

8%. 

4. 

$480 

2 yr. 5 mo. 

5%. 

14. 

$840 

3 yr. 9 mo. 12 da. 

4%. 

5. 

$540 

5 yr. 4 mo. 

7%. 

15. 

$950 

5 yr. 5 mo. 24 da. 

3%. 

6. 

$684 

2 yr. 8 mo. 

3%. 

16. 

$835 

3 yr. 10 mo. 18 da. 

7%. 

7. 

$642 

4 yr. 3 mo. 

8%. 

17. 

$785 

4 yr. 11 mo. 21 da. 

8%. 

8. 

$845 

6 yr. 5 mo. 

5%. 

18. 

$655 

3 yr. 8 mo. 27 da. 

9%. 

9. 

$920 

5 yr. 9 mo. 

4%. 

19. 

$955 

5 yr. 9 mo. 15 da. 

5%. 

10. 

$250 

3 yr. 11 mo. 

8%. 

20. 

$435 

2 yr. 7 mo. 27 da. 10%. 



Compound Interest. 


374. 

When the interest for regular intervals of time is 


added to the principal to make a new principal, such inter¬ 
est is called compound interest. 

375. The interest may be added to the principal annu¬ 
ally, semi-annually, quarterly, or at any other regular 
interval. It is understood to be added annually when no 
time is specified. 

376. The payment of compound interest cannot usually 
be enforced by law. 

377. What is the compound interest of $800 for 3 yr. 
6 mo. at 6% ? 


$800 

.06 


$898.88 

.06 


48.00 

800 


58.9328 

898.88 


$848 Amt. in 1 yr. 
.06 
50.88 
848 

$898.88 Amt. in 2yr. 


$952.81 Amt. in 3 yr. 

.06 

57.1686 

952.81 Amt. in 3 yr. 6 mo. 
$1009.98 
800 


$209.98 T nterest. Ans. . 











BANK DISCOUNT. 


207 


Find the compound interest on 
Principal. Time. Rate. 

1. $800, 3 yr. 6 mo., 6%. 

2. $168, 3 yr. 11 mo., 8%. 

3. $235, 3 yr. 6 mo., 6%. 

4. $480, 2 yr. 5 mo., 5%. 

5. $540, 5 yr. 4 mo., 7%. 

11. $850, 2 yr. 5 mo. 18da., 7%, 

12. $780, 2 yr. 8 mo. 24da., 5%, 

13. $650, 4 yr. 6 mo. 18da., 8%, 

14. $840, 3 yr. 9 mo. 12da., 4%, 

15. $950, 5 yr. 5 mo. 24da., 3%, 


each of the following : 
Principal. Time. Rate, 

6. $684, 2 yr. 8 mo., 3%. 

7. $642, 4 yr. 3 mo., 8%. 

8. $845, 6 yr. 5 mo., 5%. 

9. $920, 5 yr. 9 mo., 4%. 
10 . $250, 3 yr. 11 mo., 8%. 
compounded semi-annually, 
compounded semi-annually, 
compounded semi-annually, 
compounded quarterly, 
compounded quarterly. 


378. Bank Discount. 

When money is borrowed from persons, the interest is 
paid with the principal at the maturity of the note. 

379. When money is borrowed from a bank, the interest 
is paid on the day on which the money is borrowed. This 
simple interest, which the bank takes before it is due, is 
called bank discount. The remainder, after subtracting 
the discount, is called the proceeds. 

380. A bank calculates discount on the value of a note 
at maturity for the exact time it must hold the note. 

381. 


m 

$ 8o °- Chicago, Ill. °ct. i2, J89 -r 

in 

Ninety days after date, / promise to pay 

m 

to the order of Martin Fan Buren, 


\ ^ ^ - - - 

Eight hundred -—--- oo rv,IV* 

il 

luo 

Value Received, witb lnterest at 6%. 

11 

Nrv ^ Peter Quince. 























208 


THE 20th CENTURY ARITHMETIC. 


This note (bottom page 207) was discounted at a bank at 
10%, Nov. 11, 1892. What were the proceeds ? 

Analysis. —The value of this note at maturity is $812.40. The bank 
must wait 63 days before it receives the value of the note. The sim¬ 
ple interest on $812.40 for 63 days at 10% is $14.22. This is the bank 
discount. The proceeds are $812.40—$14.22, which is $798.18. Ans. 

Find the proceeds of the following notes : 



$ l2 5 °~ Raleigh, N. C. 7 . 1896. 

Sixty days after date, / promise to pay 
to the order of tViUiam H. Harrison, 

Twelve hundred fifty -——- oo Dollar? 

100 

Value Received, with interest at 6%. 

No. 9 % ob R-°y- 

1* Discounted Aug. 9, 1896. 

Ssk* 

$ 86 5 - Richmond, Va. Se P* 1 > \$ 9 6 - 

Five months after date, /_ promise to pay 

to the order of J°^ n Tvler, 

Eight hundred sixty-five .-- oo 

100 

Value Received, with interest at 7%. 

No. 10 Tom Tulliver. 


2. Discounted Dec. 4, 1896. 

























BANK DISCOUNT. 


209 


J 

$ 53 6 - Nashville, Term. ^ U S- 3> IK9 6 - 

SI 

Ninety days after date. / promise t0 pay 


to the order of J ames K - Polk, 


Five hundred thirty-six - - - -— 50 f)r»Hars 

SI 

100 

Value Received, interest at 8 %. 

ill 

lMn. 11 Enoch Arden. 


3. Discounted Oct. 4, 1896. 


OXC) 

y 0l s 

«o\ 

o\o 

&OX 

GSP, 

y 0l s 

CXO 1 

(5oN 

C\p. 

S oX \ 

G\p. 

<50\ 

$ 87 - Frankfort, Ky. J ulv 7 7. J89 6 

0 «« hundred days after date, / p mrn ;<;» tr> pay 
° to the order of Zachary Taylor, 

l Eig^seven - -- oo Dollars 

0 100 

l Value Received, with interest at 5*>- 

° No. 12 PWip Pay- 

4 . 

Discounted Sept. 3, 1896. 

... 

«• • 

2 $ I28 '~ Columbus. O. O ct - l6 > JgQ<5. 

_ Sixt >’ da y $ a f ter date ’l promise to pay 

* to the order of Millard Fillmore, 

IS One hundred twenty-eight - - - 65 p 0 J] aro 

• Value Received, mth interest at 7%. 

■ No. 1 3 Peter Simple. 

5. Discounted Oct. 31, 1896. 














































210 


THE 20th CENTURY ARITHMETIC. 


The following notes were made and discounted in 1896 :* 


Principal. 

Rate. 

Time. 

Date. 

Discounted 

6. 

$788.80, 

7%, 

Ninety days, 

Jan. 15, 

Feb. 29. 

7. 

$682.90, 

5%, 

Sixty days, 

Feb. 29, 

April 3. 

8. 

$1289, 

4%, 

Thirty days, 

March 16, 

March 31. 

9. 

$75, 

8 %, 

100 days, 

April 17, 

June 7. 

10. 

$681.20 

9%, 

Ninety days, 

May 18, 

July 7. 

11. 

$1841, 

64%, 

Sixty days, 

June 19, 

July 7. 

12. 

$67.52, 

74%, 

Thirty days, 

July 20, 

July 31. 

13. 

$841, 

84%, 

100 days, 

Aug. 21, 

Oct. 2. 

14. 

$2884, 

54%, 

Ninety days, 

Sept. 22, 

Oct. 31. 

15. 

$1776, 

6 f %, 

Sixty days, 

Oct. 23, 

Nov. 17. 


* 1896 was a leap year. 


382. True Discount. 

The first note under bank discount (page 207) was worth 
$800 on Oct. 12th. The interest for 80 days is $4, so that 
on Nov. 11th, the note is worth $804. Sixty days from date, 
on December 11th, the note is worth $808; at maturity it is 
worth $812.40. 

383. The difference between the actual value of a note 
and its value at maturity is called the true discount. The 
true discount of the above note on Nov. 11th is $812.40— 
$804. which is $8.40. 

384. The actual value of a note at any time before it is 
paid is called its present worth. It is that sum of money 
which, if put at interest at the current rate, will give an 
amount equal to the value of the note when it is due. 

The present worth of the above note on Nov. 11th is $804. 




TRUE DISCOUNT . 


211 


385 . 



$ l 49 2 *' i Chicago, HI .October^ j 89 + 

_ Sixty days after date, I p rnrn ^t> to pay 

to the order of _ Franklin Pierce, 

One thousand four hundred ninety-two oo Drillers 

100“ 

Value Received. 

14 Oliver Twist 


If the current rate of interest is 5%, what is the present 
worth of this note on November 1, 1892 ? 


Analysis. —Since this note does not bear interest, its value at ma¬ 
turity is its face, $1492. Its value on Nov. 1 will be a sum of money, 
which, if put at interest on Nov. 1, will amount to $1492 on the day 
the note matures, Dec. 14, 1892. 

From Nov. 1 to Dec. 14 is 44 days. $1 in 44 days at 5% will 
amount to $1,006+ How many dollars will amount to $1492 ? Evi¬ 
dently, $1492 -J-$l. 006+ which is $1482.93? The true discount is 
$1492—$1482.93, which is $9.07+. Ans. 

386. What is the present worth on the day on which it was 
discounted of each of the notes , under hank discount ? 

387. What is the discount on each of the following t 

Discounted Discounted 


Value at 
maturity. 

before 

maturity. 

Rate. 

Value at 
maturity. 

before 

maturity. Rate. 

1. $1492, 

46 days, 

5%. 

6. $1262, 

61 days, 5%. 

2. $780, 

85 days, 

5%. 

7. $2841, 

61 days, 6%. 

3 . $568, 

22 days, 

7%. 

8 . $976, 

85 days, 7%. 

4 . $184, 

19 days, 

7%. 

9 . $745, 

16 days, 7%. 

5 . $265, 

82 days, 

8%. 

10 . $858, 

28 days, 8%. 

Find the 

true discount of 

the examples under bank dis- 


count. Note the difference between the true and the bank 


discount. 














212 


THE 20th CENTURY ARITHMETIC. 


388. Partial Payments. 

A borrower often pays back the money, part at one time 
and part at another. These are called partial payments. 

The date and amount of each payment are endorsed 
on the note (written on the back). 

The amount due on a note bearing interest and having 
endorsements is to be calculated on the principle that com¬ 
pound interest is not allowed by law. 

389. The United States Rule. 


Find the amount to the time when a payment, or the sum 
of two or more payments, equals, or exceeds, the interest then due. 

Subtract from this amount the payment, or sum of payments, 
and use the remainder as a new principal. Proceed thus to the 
time of settlement. 


390. 


Face of Note. 


'a?* 

'^P 

‘UP 

‘i=hP 

'UP 


'UP 

'UP 

'UP 

'UP 

'UP 

'UP 

UP 

'UP 

‘UP 

UP 


$ ‘49^ Columbus, O. 0cL ,2 ’ .. 189^: 

- 0n dema, ? d ’ j promise to pay 

to the order o f J ames Buchanan, 

One thousand four hundred ninety-two oo Dollars 

100 

Value Received, with interest at 6%. _ 

No*. 13 Julius Caesar. 













PARTIAL PAYMENTS. 


218 


Endorsements. 


o 

8 

-Si 

■K. 

'8 

O 

■8 






6 

U's 

• 


o 


o 

o 




SO 

% 



> 

» 






<N 



<N 



a 

-si 



Q 




•s 








Os 

Os 

Os 

Os 

Oo 

oo 

Oo 

00 


*-« 




Yr. 

1892 

1892 

1898 

1898 

1894 

1894 


Find the balance due Aug. 8, 1894. 
Preliminary Work. 

Times between 
Dates. successive dates 

Mo. da. 


mo. 

10 

12 

2 

10 

7 

8 


da. 

12 

24 

21 

12 

8 

8 


Amt. due when pay¬ 
ment was made, 
1st payment, 

New principal, 


12 

27 

21 

21 

0 


Solution.— 


$1492 
.012 
2984 
1492 
17.904 
1492 

$1509.90 

100 

$1409.90 

.0095 

704950 

126891 

18.894050 


Interest on $1 
at 6 % for time 
differences. 

$.012 

.0095 

.0885 

.0485 

.005 


Payment. 

$100 
225 
40 
460 
Due ? 


$18.89 

Amt. due when 1409 90 

Paymen made! 14209 
2d payment, 225 

New principal, $1198.29 
.0885 
599145 
958682 
Payment 859487 


too small, 46.184165 















214 


THE 20th CENTURY ARITHMETIC. 


$1198.29 

.0485 

599145 


Interest, 

Amount due 
when paym’t 


859487 
479816 
j 52.125615 
<46.134 
1198.29 


was made, $1296.55 


3d and 4th pay¬ 
ments, 
New principal, 


$1296.55 

500. 

$796.55 

.005 

3.98275 


796.55 

Bal. due Aug. 3, 1894, $800.53 Ans. 


Notice in this solution that $40 paid on Oct. 12, 1893, did not pay 
the interest then due. Suppose we had proceeded as usual : 


46.13 

1198.29 

1244.42 
40. 

1204.42 


If we now calculate the interest on $1204.42, 
we would be calculating interest on interest, 
which is not allowed by the United States 
rule. No interest is allowed to the borrower 
on a payment which is less than the interest 
then due. 


391. What is due Aug. 3 , 1891/., on the following note ? 

1. Face of Note. 


MM, 

i 5000 00 

Chiron, III. 0ct ■ ,2 ’ 189 2 - 


m ' 

On demand, I p r0 mise to pay 


I to the order of 

Abraham Lincoln, 


fj Five thousand - 

- ---- - OO nnil a ^ 

.... 

■ Value Received, 

with interest at 6%. 

:::: 

if# 

i No. 16 

Napoleon ^Bonaparte. 


ill! 






























PARTIAL PAYMENTS. 


215 


Endorsements. 




>j~\ 


8 

"o 



■«9- 

s 



_ 







fN 




s 

o 

3 

Q 

<2 

o 


'■Q 






n 






On 

On 

On 

<o 

oo 

OO 

oo 

*>. 

oo 



IOOO oo 


2. Face of Note. 


New Orleans, La. S Si 1891- 


On demand, / 


promise to pay 


to the order of. 

One thousand 


Andrew. Johnson, 


oo 


Dollars 


Value Received, wlth interest a l 6% - 

pq 0 iy George McClellan. 


Endorsements. 


6 

u-\ 

*>N 

itn 

tN 

<N 

O 

^N 


<N 




- 




U~N 

ncT 


ITS 




<*> 

Si 

f> 

O 

-ci 

■a* 

■a. 


£ 





LT\ 

itn 


'O 

On 

On 

On 

On 

oo 

OO 

oo 

oo 






















216 


THE 20th CENTURY ARITHMETIC. 


Find the balance due Sept. 4, 1896. 
Preliminary Work. 

Times between Interest on $1 
Dates. successive dates, at 6 % for time 


Yr. 

mo. 

da. 

Mo. 

da. 

differences. 

Payment. 

1894 

1895 

9 

1 

10 

1 

3 

21 

$.0185 

$150. 

1895 

6 

5 

5 

4 

.025| 

253. 

1895 

11 

15 

5 

10 

.026§ 

247. 

1896 

2 

16 

8 

1 

,015£ 

350. 

1896 

9 

4 

6 

18 

.033 

Due ? 


3. 



$ 2000 ~ Richmond, Va. PO 7- 1R95- 


On demand, I promise to pay 
to the order of Ulysses S. Grant, 


Two thousand - oo f) 0 |l ars 


100 

Value Received, with inte rest at 8%. 


No. l $ Robert E. Lee. 


Endorsements: Sept. 1 , 1895, $100; Feb. 29, 1896, $500; 
June 15, 1896, $250. What was due Dec. 1 , 1896 ? 


4. 


r 

... 

>■. 

»ii4 

iiiliii 

$ So °- Columbus, O. M>7. 1894: 

On demand, I p rom ise to pay 
to the order of Rutherford B. Hayes, 

Eight hundred - oo rwt.,,* 

:: 

... 

■ nil 

... 

|yi 

.. . 

■ • ■ 

.. . 

lou U 1<tr5> 

Value Received, Wlth interest at 8%. 

No* Philip Sheridan. 






























PARTIAL PAYMENTS. 


217 


Endorsements : Dec. 3, 1894, $125; May 20, 1895, $125; 
Nov. 27, 1895, $250; June 2, 1896, $250; Nov. 1, 1896. $50. 
What was due Jan. 1, 1897 ? 


5. 


$ * 00 - Nashville, Tenn. 

On demand, 1 


Ma y 7> 1ft9 4- 
promise to pay 


to the order of 

Five hundred 


James A. Garfield, 


Value FWfwd, with interest at 


55_Dollars 


No. 


20 


Eli Whitney. 


Endorsements : Sept. 15, 1894, $50; Feb. 2, 1895, $50; 
Aug. 9, 1895, $75; Jan. 2, 1896, $75; July 8, 1896, $100. 
What was due Dec. 11, 1896? 

6 . 


$ 'too' 


Boston, Mass. J ulv z > \ 895: 


On demand, / 


promise to pay 


to the order of. 

Fifteen hundred 


Chester A. Arthur, 


oo 


Dollars 


Value Reretwd, mtb interest at 8%. _ 

No._ Robert Fulton . 


Endorsements : On this note, five payments of $200 each 
were made at intervals of four months, the first being on 
Oct. 1, 1895. What was due Jan. 1. 1897 ? 
























218 


THE 20th CENTURY ARITHMETIC . 


7. 


J 5200 ™ 


Buffalo, N. Y.— SepL_i _,—j 89^: 
On demand, I p rorn f Sg to pay 


to the order of 


Grover Cleveland, 


Five thousand two hundred 


Value Received, 


Dollars 


No* 


22 


Elias Howe. 


Endorsements: Nov. 1, 1896, $21.67; Feb. 1, 1897, $60; 
Mar. 1, 1897, $1000; July 7, 1897, $500; Nov. 8, 1897, $1500; 
What was due Jan. 1, 1898 ? 


8 . 


Ill] $ ’77(> w Washington, D. C._ 1895: 

!!|| _ Ok demand, / promise to pay 

t0 t he order of Benjamin Harrison, _ 

l!!l Seventeen hundred seventy-six —— - - Hollars 

llfijjiijji ' ’ loo 

llll Value Received, Wlth inte * est at 5^- 

fit! jv[ 0t 2 ) _ 5. F. C B. Morse. 


Endorsements : Jan. 1, 1896, $1000; Feb. 15, 1896, $100; 
Apr. 1, 1896, $75; May 15, 1896, $200; July 1, 1896, $100; 
Sept. 15, 1896, $250. What was due Nov. 1, 1896? 

































PARTIAL PAYMENTS. 


219 


9. 


'OH* 


'kk* 

§k+> 


vW*> 

-'kk* 


%3?>- Washington, D. C. J“" f ’ 22 : 1891' 

_1_ P” 7 promise to pay 

to the order of _ G k wer Cleveland, 

Seven hundred fifty - - oo Hollars 

100 

Value Received, TO?//; ?w/gm/ a l 7 X - 

js^ 0# 2 4 h aac Newton. 


Endorsements: Oct. 18, 1894, $150; Jan. 11, 1895, 
$150; Aug. 17, 1895; $150; Dec. 15, 1895, $150; Mar. 27, 
1896, $150. What was due Oct. 2, 1896. 


10 . 


1 

$ ,2 5°' w Montgomery, Ala. '■ 186'- 

On demand, I p rom ise to pay 

to the order of Horace Mann ’ 


Twelve hundred fifty -— — - oo Hollars 


100 

VaW R oreiwd, mtb interest at 8% - 


]Sj a 25 E. E. West. 


Endorsements : Sept. 15, 1862, $200; Dec. 1, 1862, $200; 
Feb. 15, 1868, $20; May 1, 1868, $280; July 15, 1868,$200. 
What was due Oct. 1, 1868 ? 



























220 


THE 20th CENTURY ARITHMETIC. 


392- INVOLUTION. 

A power is the product when the same factor is repeated. 

The 1st power of 5 is 5. 

The 2d power of 5 is 5 2 , or 5x5. 

The 3d power of 5 is 5 3 ,or 5x5x5. 

The 4th power of 5 is 5 4 , or 5X5X5X5, etc. 

The small figure placed above and to the right to show 
how many times the factor is to be repeated, is called an 
exponent. 

The process of finding a power is called involution. It is 
simply multiplication, using the same multiplier as often 
as is indicated by the exponent. 

The 2d power of a number is called its square. 

The 3d power of a number is called its cube. 

1. Write the squares of all the digits. 

2. Write the cubes of all the digits. 


Find the value of the following : 


3 . 

2845 2 . 

8 . 

.434 2 . 

13 . 

365 3 . 

18 . 

.0016 3 . 

23 . 

9 4 

4 . 

1492 2 . 

9 . 

.0141 2 . 

14 . 

.5 3 . 

19 . 

(4) 3 - 

24 . 

7 5 

5 . 

365 2 . 

10 . 

.0016 2 . 

15 . 

.0416 3 . 

20 . 

(#) 2 - 

25 . 

8 4 

6 . 

863 2 . 

11 . 

35 3 . 

16 . 

.434 3 . 

21 . 

(J) 3 . 

26 . 

6 5 

7 . 

3.1416 2 . 

12 . 

65 3 . 

17 . 

.0141 3 . 

22 . 

(I) 2 - 

27 . 

2 8 


393. EVOLUTION. 

The root of a number is one of its equal factors. 
The square root of a number is one of the two equal factors which 
produce it. 8X8=64 ; the square root of 64 is 8. 

The cube root of a number is one of the three equal factors which 
produce it. 3X3X3=27; the cube root of 27 is 3. 

Evolution is the process of finding the roots of numbers, 
l/ is the sign for evolution ; it is called the radical sign. 
The root to be found is indicated by a little figure called the 
index of the root. The index is omitted in square root, 
l/=square root. 1 ? / =fourth root. 

# / =cube root. ^=fifth root. 




SQUARE ROOT. 


221 


394. Square Root. 

The area of a square is found by multiplying the length 
of a side by itself (Art. 227, a). The reverse process, of find¬ 
ing the length of a side when the area is given, is now to be 
shown. 

The square root of a number is one of the two equal 
factors which produce it. 

If a number expresses the area of a square, the square root is the 
length of a side of the square. 

Number. 1 4 9 16 25 86 49 64 81 

Square root. 128456 7 89 











THE goth CENTURY ARITHMETIC. 


222 


The line AD is 158 inches long and the square on AD con¬ 
tains 158X158 sq. in.=24964 sq. in. This square contains 
the following areas : 

100 2 +2 X100X 50+50 2 +2 X150 X 8+ S ' 2 . 

=100 2 + (2X100+50)50+ (2x150+8)8=24964. 

The square of any other number may be similarly 
expressed : (Make a diagram similar to the one above.) 

1492 2 =1000 2 + ( 2X1000+400 )400+ ( 2X1400+90 )90+ 
(2X 1490+2)2=2226064. 

The reverse process is performed in extracting the square 
root. 

Examine the following: 

a. Extract the square root of 24964. 


1 st. 2'49'64|_ 


2d. 

2'49'64|1 

1 

149 

3d. 

2'49'64 11 


1 

2 

|149 

4th. 

2'49'64|15 


1 

25 1149 


5th. 2'49'64 1158 

1 


25|149 
125 


First. Beginning at units place, point 
off into periods of two figures each. 

Second. Extract square root of left 
period. Subtract square of first root 
figure, 1, from left period. Annex to 
the remainder the next period, 49. 

Third. Multiply the root already 
found by 2. Place the product, 2, as 
shown. This is the trial divisor. 

Fourth. The trial divisor, 2, is con¬ 
tained in 14 (neglect the right figure, 
9) 5 times. Annex this to the root and 
to the trial divisor. 


Fifth. Multiply the complete divisor, 
25, by the last root figure. Proceed 
with the remainder as in Third. 


808|2464 
2464 














SQUARE ROOT. 


223 


Examine these solutions: 


j/2226064 
2'22W64|1492 

1 

241122 
96 

28912660 
2601 

298215964 
5964 


|/5322249 

5 / 32'22'49| 2807 
4 

431132 
129 

4607132249 
32249 


Find the square 

1. 1369. 

2. 32768. 

3. 6561. 

4 . 1089. 

5. 1296. 

6. 8836. 

7. 6889. 

8. 11881. 

9. 14161. 

10. 156025. 


of the following : 

11. 2226064. 

12. 192721. 

13 . 444889. 

14 . 514089. 

15 . 29855296. 

16 . 5322249. 

17 . 529475129. 

18 . 89661961. 

19 . 1827904. 

20. 3154176. 


21. 39037504. 

22. 31189791. 

23 . 21224449. 

24 . 3486784401. 

25 . 82264900. 

26 . 3226694416. 

27 . 4064572516. 

28 . 282429536481 

29 . 45166720770. 
30 6407522209. 


395. Point off numbers containing decimals from the 
decimal point to the right and left in periods of two figures 
6ach. 

;/15432.3488 1st step, 1'54'82.34'88 

j/61027.05152 1st step, 6'10'27.05'15'2 

T /.03937 1st step, .03'93'7 

Begin on the left, as in integers, to extract the root. 












224 


THE 20th CENTURY ARITHMETIC. 


Examine the following: 


a. ] 3.14159 b. j/5 to 3 decimals. 

8.14'15'9|1.772+ 5.|2.236+ 

1 4 


27|214 

189 

347(2515 

2429 


421100 
84 

44311600 
1329 


354218690 
7084 


4466127100 
26796 


Extract the square root of (3 decimals) 


l. 3.14159 

6 . 16482.3489 

11- 5. 

16 . 

.3 

2. .03937 

7 . 61027.05162 

12. 3. 

17 . 

.5 

3 . .00111 

8 . 632.2249 

13 . 2. 

18 . 

.2 

4 . .129 

9 . 847.269 

14 . 7. 

19 . 

.7 

5 . 2.262 

10. .00716 

15 . 6. 

20. 

.6 


396. When both terms of a fraction are perfect squares, 
extract the square root of each term ; otherwise, reduce the 
fraction to a decimal, then extract the root. 


a - V^ 15 —b. |/|=|/.375=.612-(-. 

Extract the square root of 

1* 4. f. 7 . f. 10 . 3|. 13 . 87f. 16 . 48}. 

2 * B- 5. }. 8. f. 11. 4f. 14. 33}. 17. 16}. 

3 - ft- 6- #• 9 . f. 12 . 6|. 15 . 91}. 18 . 87}. 

397. When the second and third terms of a proportion 
are the same, it is called a mean proportional. In the pro¬ 
portion 4 : x : : x : 9, x is a mean proportional between 4 and 
9- z 2 =36 (Art. 302). Extracting the square root of both 
sides of the equation, we have a:=6. 

a. Find a mean proportional between 3 and 6. 

First step, 3: x\:x: 6. Second step, x 2 =18. Third step, a?=|/jg7 











SQUARE ROOT. 


225 


Find a mean proportional between 

1. 4 and 9. 4 . 3 and 6. 7 . 12 and 33. 10 . 91 and 78. 

2. 4 and 16. 5 . 5 and 11. 8. 21 and 19. 11 . 82 and 96. 

3. 3 and 27. 6. 7 and 16. 9 . 65 and 44. 12 . 69 and 100. 

AB is the hypotenuse. 

Examine the figure 
and note the following 
facts : First , the little 
squares are all the same 
size. Second , the 
square on the altitude 
is composed of 9 little 
.squares; on the base, 
16 little squares; and 
on the hypotenuse, 25 
little squares. 

25=16+9 ; there¬ 
fore, the square on the 
hypotenuse of a right tri¬ 
angle equals the sum of 
the squares on the other 
two sides. 


The base of a triangle is 8 inches, the altitude is 6 inches. 
What is the hypotenuse ? 

Analysis.— The square on the hypotenuse is 8 2 -f-6 2 =64-f- 36= 


100. The hypotenuse is, therefore, |/100=10. Ans. 

Find the hypotenuse of the following right triangles : 


Base. 

Altitude. 

Base. 

Altitude. 

Base. 

Altitude. 

1 . 21 

20 . 

5 . Si 

11 . 

9 . 19 

13. 

2. 45 

28. 

6 . 17 

17. 

10. 82 

50. 

3 . 45 

24. 

7 . 25 

21 . 

11 . 63 

25. 

4 . 24 

10 . 

8 . 63 

44. 

12. 41 

33. 





















226 


THE 20th CENTURY ARITHMETIC. 


399. Cube Root. 

The cube root of a number is-one of the three equal fac¬ 
tors which produce it. 

^125 is read “the cube root of 125.” |X125=5, because 5 is one 
of the three equal factors which produce 125. 

Number. 1 8 27 64 125 216 848 512 729 

Cube root. 128456789 

Note. —Let the teacher, using blocks, explain cube root similarly 
to the explanation given for square root in Art. 394. This explana¬ 
tion cannot be satisfactorily made without blocks, therefore the 
author has omitted it. 

Examine the following: 

a. Extract the cube root of 105154048. 


105'154'048| 


First. Beginning at units pjace, point off 
into periods of three figures each- 


105'154'048|4 

64 


4800 


41154 

105'154'048|4 


64 


4800 


41154 

105'154'048|47 


64 

'41154 


Second. Extract cube root of left period. 
Subtract cube of first root figure, 4, from 
left period. Annex to the remainder, the 
next period, 154. 

Third. Square the root already found, 16 ; 
multiply this by 3; annex two ciphers and 
place the result, 4800, as shown. This is the 
trial divisor. 

Fourth. Using the trial divisor we obtain 
the next figure of the root, 7. 


4800 

840 

49 

5689 


105 / 154'048|47 

64 

41154 


is the complete divisor. 


Fifth. Add to the trial divisor three times 
the product of the last root figure and the 
root already found, 4x7=28, 8X28=84; 
to this annex one cipher. Add, also, the 
square of the last root figure, 49. The sum 
Proceed with the remainder as in Third. 
















CUBE ROOT. 


227 


4 2 X3, 

4800 

105'154'048|472 

4X7X8, 

840 

64 

7 2 , 

49 

41154 


5689 

39823 

47 2 X8, 

662700 

1331048 

47X2X8, 

2820 


2», 

4 

1331048 


665524 


400. The fourth and fifth steps are the important ones. 
They may be easily remembered, as follows : 

Let a —the root already found, and 6 = the last figure of 
the root. The trial divisor is 3a 2 and the complete divisor 

is 3a 2 -f-3a6-f~& 2 - 

Examine the following solutions : 
b. Extract the cube root of 3321287488. 


a=1 , 

3a 2 , 

300 

3'321'287'488| 1492 

6=4, 

3 ah, 

120 

1 

» 


6 2 , 

16 

2321 



“436 

1744 

a=14, 

3 a 2 , 
Sab, 

58800 

577287 

6= 9, 

3780 




81 

563949 



62661 


a=149, 

3a 2 , 
Sab , 

6660300 

• 13338488 

b= 2 , 

8940 



6 2 , 


4 

13338488 



6669244 


e. Extract 

the cube root of 12278428443. 

a— 2 , 

3a 2 , 

Sab, 

1200 

12'278'428'448|2307 

b= 3, 

180 

8 


b 2 , 

9 

4278 



1389 

4167 

a=230, 

3a 2 , 

15870000 

111428443 

7, 

Sab, 

48300 



b 2 , 

49 

111428443 


15918349 























228 


THE 20th CENTURY ARITHMETIC. 


401. The trial divisor may be easily obtained from the 
preceding work, as follows : Write the value of b 2 under the 
last complete divisor and add the three numbers above to 
it; annex two ciphers to the sum. 

In a. 


Trial divisor 


840 

In b. 120 

8780 

49 

16 

81 

5689 

486 

62661 

49 

16 

81 

662700 

58800 

6660300 


402. Point off numbers containing decimals to the right 
and left from the decimal point into periods of three figures 
each. 


Unless both terms of a common fraction are perfect cubes, 
reduce the fraction to a decimal before extracting the root. 


Find the cube root of the following: 


1. 912673. 

2. 24389. 

3 . 148877. 

4 . 8365427. 

5 . 14706125. 

6. 12812904. 

7 . 84027672. 

8. 529475129. 

9 . 1291467969. 
10. 233744896. 


11. 134217728. 

12. 27054.086008. 

13 . 4516672.077. 

14 . 102503.232. 

15 . 355496768704. 

16 . .024137569. 

17 . 13841287201. 

18 . .007821346625. 

19 . 59.776471. 

20. .056711623688. 


21. 176464.081529. 

22. 2.222447625. 

23 . .081690010219. 

24 . 198767717056. 

25 . 8452.26465300. 

26 . 3.141592653589. 

27 . 34829 (3 decimals). 

28 . .30103 (4decimals). 

29 . 7 (5 decimals). 

30 . .09 (5 decimals). 


4 ° 3 - FORMULAS. 

Take any two numbers; for example, 9 and 4. Their sum is 13 
Their difference is 5. 13+5=18 ; * of 18=9, the larger number. ' 

Take any two numbers : 

4 ( their Bum their difference) =the larger number. 







CUBE ROOT. 


229 


Prove this to be true for the numbers 7 and 2 ; 9 and 8 ; 
12 and 5 ; 25 and 10 ; 86 and 20. When a general statement 
like the above is to be made in mathematical language, 
numbers are represented by letters of the alphabet. 

Let a represent any number and b any other number ; 
then, a+b is their sum ; and a—b is their difference. Ex¬ 
pressed in mathematical language, the statement above 
becomes \ (a-\-b-\-a — b)=a. 

404. A mathematical formula is a truth represented by 
means of symbols and letters. 

405. The symbols of operation, -J-, —, X, j/ , etc., 
are used with letters just as with numbers. The multipli¬ 
cation sign, however, is usually omitted between letters. 

aXfr may also be written ab. xyz means zX^/Xz* 

406. A coefficient is a number which shows how many 
times the quantity is used as an addendum. 
x-\-x~\-x-^-x=4 x\ 4 is the coefficient. 

407. An exponent is a number which shows how many 
times the quantity is used as a factor. 

«X«Xct is written a 3 3 is the exponent. 

408. If a and b represent two numbers, what will repre¬ 
sent 

1 . Their sum ? 5. The square of their sum ? 

2. Their difference ? 6. The sum of their squares ? 

3. Their product ? 7. The product of their sum and 

4. Their quotient ? difference ? 

8. Answer the same questions for the letters x and y ; m 
and n ; b and c ; r and s. 

409. Find the value of a 1 ^ when a= 9, b= 5, and c= 2. 

c 

Solution.- ^±j. = 9 + 5 =7 Am . 
c 2 




280 


THE 20th CENTURY ARITHMETIC. 


Find the value of the following when a=9, b= 5, and c=2 : 


9 . a+6+c 3 . 

10. 8a— b 2 -j-c. 

11. 5 a —26 2 +c 3 . 

12. 7a 2 — 86 2 +9c 4 
feet of the side of a 


1. a+fr+c. 5. 2a-\-b-\-c. 

2. a+6—c. 6. a+26—8c. 

3 . a—6+c. 7 . 8a— 2b —3c. 

4 . a — b — c. 8 . 26+2c — a. 

13 . If a represents the length 
square, what is its area ? 

14. If the side of a rectangle is a feet long and its base b 
feet long, what is its area ? 

15. What is the volume of a cube whose sides are each a 
feet long ? 

16. Show that a x +a and a x — a are even 

a. When a—5 and x=S. b. When a= 6 and x=2. 

C. When a= 7 and x=S. 


17 . Show that n 5 —n is divisible by 30 when n—4. 

18 . Show that n 2 —1 is divisible by 24 when n— 17. 

19 . Show that n(n-fl)(2n-f 1 ) is divisible by 6 when n—8. 

20. Show that (n 2 +3) (w 2 + 7) is divisible by 32 when n is 
odd. Take n= 5. 


410. The Circle. 

The circumference of every circle is 3.1416+ times as long 
as its diameter. This important mathematical constant is 
represented by the Greek letter, tt. Its value is about 3+ 
or more nearly 

Let r represent the radius of a circle. 

The circumference of a circle=2 Trr. 

The area of a circle=7r r 2 . 

a. Find the circumference and area of a circle whose ra¬ 
dius is 7 inches. 

Solution. 

Cir. =27r r=2 X3.1416x7=48.9824 inches. Ans. 

Area = 7rr 2 =3.1416x49=153.9384 sq. in. Ans. 



THE SPHERE. 


281 


Find the circumference and area of a circle whose radius is 

1. 8 in. 3. 10 in. 5. 18 in. 7. 7 ft. 9. 25 yd. 

2. 5 in. 4. 16 in. 6. 2 ft. 8. 19 yd. 10. 1 mi. 


411. The Sphere. 

Let r represent the radius of a sphere. 

The surface of a sphere=4 re r 2 . 

The volume of a sphere=f7rr 3 . 

Find the surface and volume of spheres having the radii 
used in Art. 410. 


412. Miscellaneous Examples. 

1. The expression x 2 H-z+41 gives prime numbers when 
the value of x is 10 or less. Prove it for 2, 8, 5, and 8. 

2. 1+8+5-F7+ .... +(271— l)=n 2 . a. Provethisto be 
true when n— 7. b. When ti=10. C. When n= 12. 

3. If the base of a triangle is b and its height is h, its 
area is £ bh. a. What is the area of a triangle whose base is 9 
in. and height 6 in.? b. 6=11, h=8. C. 6=12, h— 5. 

4. Falling Bodies. </=82.2, s=space fallen in feet, 
£=time of fall in seconds, ^—velocity acquired in feet per 
second. 

_ 2 s v 2 

<i )s=igt 2 . ( 2 )v=gt. (z)v=y / 2gs. (5)s — 

a. What is the time occupied by a body in falling from a 
height of 750 feet, and what is the velocity with which it 
strikes the ground ? Ans. t— 6.8 sec. t;= 219.8 ft. per sec. 

b. From what height must a body fall to acquire a veloc¬ 
ity of 1500 feet per sec.? Use (&). 




282 


THE 20th CENTURY ARITHMETIC. 


C. How long would a stone be in falling to the bottom of 
a well 1 mile deep ? Use ( 4 ). 

d. How far will a bullet fall in 5 seconds ? Use 0). 

What will be its velocity ? Use ( 2 ). 

5 . Interest. P=principal, R=rate per cent, T=time 
in years, I=interest. 


(i) PRT=I. ( 2 ) P =J- 


(3) T; 


i. ( 4 )R= J_. 

RT' ‘ PR' PT 

a. Solve the examples in Arts. 858,355, and 357, using these 
formulas. 


413. 


Progressions. 


An arithmetical progression is a series of numbers 

which increase or decrease by a common difference. 

1, 3, 5, 7, 9, 11, 13. 20, 16, 12, 8, 4. 

A geometrical progression is a series of numbers, each 
of which is obtained by multiplying the preceding number 
by a constant number, called the ratio. 

1, 3, 9, 27, 81, 248. 64, 32, 16, 8, 4, 2, 1. 


In these progressions, let <z=the first term, l=the last term, n= 
the number of terms, s=the sum of the terms, d=the common dif¬ 
ference, and r=the ratio. 


Arithmetical Progression. 


l=a-\- ( n—1 )d. 


I—a 

n =~r+ 1- 



_ 2s 7_ l a 

n— -• d= -. 

a~\~l n —1 


1. 


a=l 
d=2 
n =7 


8 =? 


Geometrical Progression. 


1= 

=ar n ~ 1 . 

a (»•“->) 

l 

ci — 


r _1 ‘ 

r n-i 

l- 

a~\- ( r —l)s 

rl—a 

,9- 

*—<a 


r 

r —1 ' 

s—l * 


2 . 


a =1 
r=3 

n=z 9 


l=? 

s=? 





















